Theorem bj-omex2 15950

Description: Using bounded set induction and the strong axiom of infinity, om is a set, that is, we recover ax-infvn 15914 (see bj-2inf 15911 for the equivalence of the latter with bj-omex 15915). (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 15949	. . 3 |- E. a A. x ( x e. a <-> ( x = (/) \/ E. y e. a x = suc y ) )
2		vex 2775	. . . 4 |- a e. _V
3		bdcv 15821	. . . . 5 |- BOUNDED a
4	3	bj-inf2vn 15947	. . . 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 1625	. 2 |- E. a a = om
7	6	issetri 2781	1 |- om e. _V

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 710   A.wal 1371   = wceq 1373   e. wcel 2176   E.wrex 2485   _Vcvv 2772   (/)c0 3460   suc csuc 4413   omcom 4639

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-nul 4171  ax-pr 4254  ax-un 4481  ax-bd0 15786  ax-bdim 15787  ax-bdor 15789  ax-bdex 15792  ax-bdeq 15793  ax-bdel 15794  ax-bdsb 15795  ax-bdsep 15857  ax-bdsetind 15941  ax-inf2 15949

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-suc 4419  df-iom 4640  df-bdc 15814  df-bj-ind 15900

This theorem is referenced by: (None)