Theorem bj-omex2 16340

Description: Using bounded set induction and the strong axiom of infinity, om is a set, that is, we recover ax-infvn 16304 (see bj-2inf 16301 for the equivalence of the latter with bj-omex 16305). (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 16339	. . 3 |- E. a A. x ( x e. a <-> ( x = (/) \/ E. y e. a x = suc y ) )
2		vex 2802	. . . 4 |- a e. _V
3		bdcv 16211	. . . . 5 |- BOUNDED a
4	3	bj-inf2vn 16337	. . . 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 1648	. 2 |- E. a a = om
7	6	issetri 2809	1 |- om e. _V

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 713   A.wal 1393   = wceq 1395   e. wcel 2200   E.wrex 2509   _Vcvv 2799   (/)c0 3491   suc csuc 4456   omcom 4682

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-nul 4210  ax-pr 4293  ax-un 4524  ax-bd0 16176  ax-bdim 16177  ax-bdor 16179  ax-bdex 16182  ax-bdeq 16183  ax-bdel 16184  ax-bdsb 16185  ax-bdsep 16247  ax-bdsetind 16331  ax-inf2 16339

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-pr 3673  df-uni 3889  df-int 3924  df-suc 4462  df-iom 4683  df-bdc 16204  df-bj-ind 16290

This theorem is referenced by: (None)