per cases
( Y = {} or Y <> {} )
;

end;

suppose
Y <> {}
; :: thesis: not PARTITIONS Y is empty

then reconsider Y = Y as non empty set ;

A1: ( {Y} is Subset-Family of Y & union {Y} = Y ) by ZFMISC_1:25, ZFMISC_1:68;

for A being Subset of Y st A in {Y} holds

( A <> {} & ( for B being Subset of Y holds

( not B in {Y} or A = B or A misses B ) ) )

hence not PARTITIONS Y is empty by Def3; :: thesis: verum

end;A1: ( {Y} is Subset-Family of Y & union {Y} = Y ) by ZFMISC_1:25, ZFMISC_1:68;

for A being Subset of Y st A in {Y} holds

( A <> {} & ( for B being Subset of Y holds

( not B in {Y} or A = B or A misses B ) ) )

proof

then
{Y} is a_partition of Y
by A1, EQREL_1:def 4;
let A be Subset of Y; :: thesis: ( A in {Y} implies ( A <> {} & ( for B being Subset of Y holds

( not B in {Y} or A = B or A misses B ) ) ) )

assume A2: A in {Y} ; :: thesis: ( A <> {} & ( for B being Subset of Y holds

( not B in {Y} or A = B or A misses B ) ) )

then A3: A = Y by TARSKI:def 1;

thus A <> {} by A2, TARSKI:def 1; :: thesis: for B being Subset of Y holds

( not B in {Y} or A = B or A misses B )

let B be Subset of Y; :: thesis: ( not B in {Y} or A = B or A misses B )

assume B in {Y} ; :: thesis: ( A = B or A misses B )

hence ( A = B or A misses B ) by A3, TARSKI:def 1; :: thesis: verum

end;( not B in {Y} or A = B or A misses B ) ) ) )

assume A2: A in {Y} ; :: thesis: ( A <> {} & ( for B being Subset of Y holds

( not B in {Y} or A = B or A misses B ) ) )

then A3: A = Y by TARSKI:def 1;

thus A <> {} by A2, TARSKI:def 1; :: thesis: for B being Subset of Y holds

( not B in {Y} or A = B or A misses B )

let B be Subset of Y; :: thesis: ( not B in {Y} or A = B or A misses B )

assume B in {Y} ; :: thesis: ( A = B or A misses B )

hence ( A = B or A misses B ) by A3, TARSKI:def 1; :: thesis: verum

hence not PARTITIONS Y is empty by Def3; :: thesis: verum