Metamath Proof Explorer


Theorem ominf

Description: The set of natural numbers is infinite. Corollary 6D(b) of Enderton p. 136. (Contributed by NM, 2-Jun-1998) Avoid ax-pow . (Revised by BTernaryTau, 2-Jan-2025)

Ref Expression
Assertion ominf ¬ ω Fin

Proof

Step Hyp Ref Expression
1 isfi ω Fin x ω ω x
2 nnord x ω Ord x
3 ordom Ord ω
4 ordelssne Ord x Ord ω x ω x ω x ω
5 2 3 4 sylancl x ω x ω x ω x ω
6 5 ibi x ω x ω x ω
7 df-pss x ω x ω x ω
8 6 7 sylibr x ω x ω
9 nnfi x ω x Fin
10 ensymfib x Fin x ω ω x
11 9 10 syl x ω x ω ω x
12 11 biimpar x ω ω x x ω
13 pssinf x ω x ω ¬ ω Fin
14 8 12 13 syl2an2r x ω ω x ¬ ω Fin
15 14 rexlimiva x ω ω x ¬ ω Fin
16 1 15 sylbi ω Fin ¬ ω Fin
17 pm2.01 ω Fin ¬ ω Fin ¬ ω Fin
18 16 17 ax-mp ¬ ω Fin