Theorem bj-inf2vn 15184

Description: A sufficient condition for om to be a set. See bj-inf2vn2 15185 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-inf2vn.1	|- BOUNDED A

Assertion

Ref	Expression
bj-inf2vn	|- ( A e. V -> ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A = om ) )

Distinct variable group:   x, y, A

Allowed substitution hints:   V( x, y)

Proof of Theorem bj-inf2vn

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-inf2vnlem1 15180	. . 3 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> Ind A )
2		biimp 118	. . . . . . 7 |- ( ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> ( x e. A -> ( x = (/) \/ E. y e. A x = suc y ) ) )
3	2	alimi 1466	. . . . . 6 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A. x ( x e. A -> ( x = (/) \/ E. y e. A x = suc y ) ) )
4		df-ral 2473	. . . . . 6 |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) <-> A. x ( x e. A -> ( x = (/) \/ E. y e. A x = suc y ) ) )
5	3, 4	sylibr 134	. . . . 5 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A. x e. A ( x = (/) \/ E. y e. A x = suc y ) )
6		bj-inf2vn.1	. . . . . 6 |- BOUNDED A
7		bdcv 15058	. . . . . 6 |- BOUNDED z
8	6, 7	bj-inf2vnlem3 15182	. . . . 5 |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind z -> A C_ z ) )
9	5, 8	syl 14	. . . 4 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> (Ind z -> A C_ z ) )
10	9	alrimiv 1885	. . 3 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A. z (Ind z -> A C_ z ) )
11	1, 10	jca 306	. 2 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> (Ind A /\ A. z (Ind z -> A C_ z ) ) )
12		bj-om 15147	. 2 |- ( A e. V -> ( A = om <-> (Ind A /\ A. z (Ind z -> A C_ z ) ) ) )
13	11, 12	imbitrrid 156	1 |- ( A e. V -> ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A = om ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   A.wal 1362   = wceq 1364   e. wcel 2160   A.wral 2468   E.wrex 2469   C_ wss 3144   (/)c0 3437   suc csuc 4383   omcom 4607  BOUNDED wbdc 15050  Ind wind 15136

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-nul 4144  ax-pr 4227  ax-un 4451  ax-bd0 15023  ax-bdim 15024  ax-bdor 15026  ax-bdex 15029  ax-bdeq 15030  ax-bdel 15031  ax-bdsb 15032  ax-bdsep 15094  ax-bdsetind 15178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-suc 4389  df-iom 4608  df-bdc 15051  df-bj-ind 15137

This theorem is referenced by:  bj-omex2  15187  bj-nn0sucALT  15188