Theorem bj-inf2vn 13172

Description: A sufficient condition for om to be a set. See bj-inf2vn2 13173 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 13168	. . 3 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> Ind A )
2		bi1 117	. . . . . . 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 1431	. . . . . 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 2421	. . . . . 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 133	. . . . 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 13046	. . . . . 6 |- BOUNDED z
8	6, 7	bj-inf2vnlem3 13170	. . . . 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 1846	. . 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 304	. 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 13135	. 2 |- ( A e. V -> ( A = om <-> (Ind A /\ A. z (Ind z -> A C_ z ) ) ) )
13	11, 12	syl5ibr 155	1 |- ( A e. V -> ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A = om ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   A.wal 1329   = wceq 1331   e. wcel 1480   A.wral 2416   E.wrex 2417   C_ wss 3071   (/)c0 3363   suc csuc 4287   omcom 4504  BOUNDED wbdc 13038  Ind wind 13124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054  ax-pr 4131  ax-un 4355  ax-bd0 13011  ax-bdim 13012  ax-bdor 13014  ax-bdex 13017  ax-bdeq 13018  ax-bdel 13019  ax-bdsb 13020  ax-bdsep 13082  ax-bdsetind 13166

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505  df-bdc 13039  df-bj-ind 13125

This theorem is referenced by:  bj-omex2  13175  bj-nn0sucALT  13176