Theorem bj-inf2vn 16914

Description: A sufficient condition for om to be a set. See bj-inf2vn2 16915 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 16910	. . 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 1508	. . . . . 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 2533	. . . . . 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 16788	. . . . . 6 |- BOUNDED z
8	6, 7	bj-inf2vnlem3 16912	. . . . 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 1927	. . 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 16877	. 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 720   A.wal 1400   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   C_ wss 3220   (/)c0 3520   suc csuc 4505   omcom 4732  BOUNDED wbdc 16780  Ind wind 16866

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-nul 4254  ax-pr 4341  ax-un 4573  ax-bd0 16753  ax-bdim 16754  ax-bdor 16756  ax-bdex 16759  ax-bdeq 16760  ax-bdel 16761  ax-bdsep 16824  ax-bdsetind 16908

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-suc 4511  df-iom 4733  df-bdc 16781  df-bj-ind 16867

This theorem is referenced by:  bj-omex2  16917  bj-nn0sucALT  16918