Theorem bj-omex2 16764

Description: Using bounded set induction and the strong axiom of infinity, om is a set, that is, we recover ax-infvn 16728 (see bj-2inf 16725 for the equivalence of the latter with bj-omex 16729). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-omex2	|- om e. _V

Proof of Theorem bj-omex2

Dummy variables x y a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-inf2 16763	. . 3 |- E. a A. x ( x e. a <-> ( x = (/) \/ E. y e. a x = suc y ) )
2		vex 2818	. . . 4 |- a e. _V
3		bdcv 16635	. . . . 5 |- BOUNDED a
4	3	bj-inf2vn 16761	. . . 4 |- ( a e. _V -> ( A. x ( x e. a <-> ( x = (/) \/ E. y e. a x = suc y ) ) -> a = om ) )
5	2, 4	ax-mp 5	. . 3 |- ( A. x ( x e. a <-> ( x = (/) \/ E. y e. a x = suc y ) ) -> a = om )
6	1, 5	eximii 1651	. 2 |- E. a a = om
7	6	issetri 2825	1 |- om e. _V

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 716   A.wal 1396   = wceq 1398   e. wcel 2205   E.wrex 2523   _Vcvv 2815   (/)c0 3510   suc csuc 4488   omcom 4714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-nul 4238  ax-pr 4324  ax-un 4556  ax-bd0 16600  ax-bdim 16601  ax-bdor 16603  ax-bdex 16606  ax-bdeq 16607  ax-bdel 16608  ax-bdsb 16609  ax-bdsep 16671  ax-bdsetind 16755  ax-inf2 16763

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-sn 3697  df-pr 3698  df-uni 3917  df-int 3952  df-suc 4494  df-iom 4715  df-bdc 16628  df-bj-ind 16714

This theorem is referenced by: (None)