Theorem bj-inf2vn2 16762

Description: A sufficient condition for om to be a set; unbounded version of bj-inf2vn 16761. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-inf2vn2	|- ( 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-inf2vn2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-inf2vnlem1 16757	. . 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 1504	. . . . . 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 2527	. . . . . 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-inf2vnlem4 16760	. . . . 5 |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind z -> A C_ z ) )
7	5, 6	syl 14	. . . 4 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> (Ind z -> A C_ z ) )
8	7	alrimiv 1923	. . 3 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A. z (Ind z -> A C_ z ) )
9	1, 8	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 ) ) )
10		bj-om 16724	. 2 |- ( A e. V -> ( A = om <-> (Ind A /\ A. z (Ind z -> A C_ z ) ) ) )
11	9, 10	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 716   A.wal 1396   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   C_ wss 3213   (/)c0 3510   suc csuc 4488   omcom 4714  Ind wind 16713

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-nul 4238  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-bd0 16600  ax-bdor 16603  ax-bdex 16606  ax-bdeq 16607  ax-bdel 16608  ax-bdsb 16609  ax-bdsep 16671

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-sn 3697  df-pr 3698  df-uni 3917  df-int 3952  df-suc 4494  df-iom 4715  df-bdc 16628  df-bj-ind 16714

This theorem is referenced by: (None)