Theorem bj-omex2 14611

Description: Using bounded set induction and the strong axiom of infinity, om is a set, that is, we recover ax-infvn 14575 (see bj-2inf 14572 for the equivalence of the latter with bj-omex 14576). (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 14610	. . 3 |- E. a A. x ( x e. a <-> ( x = (/) \/ E. y e. a x = suc y ) )
2		vex 2740	. . . 4 |- a e. _V
3		bdcv 14482	. . . . 5 |- BOUNDED a
4	3	bj-inf2vn 14608	. . . 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 1602	. 2 |- E. a a = om
7	6	issetri 2746	1 |- om e. _V

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 708   A.wal 1351   = wceq 1353   e. wcel 2148   E.wrex 2456   _Vcvv 2737   (/)c0 3422   suc csuc 4365   omcom 4589

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4129  ax-pr 4209  ax-un 4433  ax-bd0 14447  ax-bdim 14448  ax-bdor 14450  ax-bdex 14453  ax-bdeq 14454  ax-bdel 14455  ax-bdsb 14456  ax-bdsep 14518  ax-bdsetind 14602  ax-inf2 14610

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-suc 4371  df-iom 4590  df-bdc 14475  df-bj-ind 14561

This theorem is referenced by: (None)