A1: {2,3,5} c= NAT

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {2,3,5} or x in NAT )
assume x in {2,3,5} ; :: thesis: x in NAT
then ( x = 2 or x = 3 or x = 5 ) by ENUMSET1:def 1;
hence x in NAT ; :: thesis: verum

end;

for q being Nat holds
( q in {2,3,5} iff ( q < 7 & q is prime ) )

proof

let q be Nat; :: thesis: ( q in {2,3,5} iff ( q < 7 & q is prime ) )

hereby :: thesis: ( q < 7 & q is prime implies q in {2,3,5} )

assume q in {2,3,5} ; :: thesis: ( q < 7 & q is prime )
then ( q = 2 or q = 3 or q = 5 ) by ENUMSET1:def 1;
hence ( q < 7 & q is prime ) by INT_2:28, PEPIN:41, PEPIN:59; :: thesis: verum

end;

assume Z: ( q < 7 & q is prime ) ; :: thesis: q in {2,3,5}
then q < 6 + 1 ;
then q <= 6 by NAT_1:13;
then not not q = 0 & ... & not q = 6 ;
hence q in {2,3,5} by ENUMSET1:def 1, Z, INT_2:29, lem6; :: thesis: verum

end;

hence SetPrimenumber 7 = {2,3,5} by A1, NEWTON:def 7; :: thesis: verum