Theorem bj-inf2vn2 14888

Description: A sufficient condition for om to be a set; unbounded version of bj-inf2vn 14887. (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 14883	. . 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 1455	. . . . . 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 2460	. . . . . 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 14886	. . . . 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 1874	. . 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 14850	. 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 708   A.wal 1351   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   C_ wss 3131   (/)c0 3424   suc csuc 4367   omcom 4591  Ind wind 14839

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 4131  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-bd0 14726  ax-bdor 14729  ax-bdex 14732  ax-bdeq 14733  ax-bdel 14734  ax-bdsb 14735  ax-bdsep 14797

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 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-suc 4373  df-iom 4592  df-bdc 14754  df-bj-ind 14840

This theorem is referenced by: (None)