Theorem bj-nn0suc0 16720

Description: Constructive proof of a variant of nn0suc 4726. For a constructive proof of nn0suc 4726, see bj-nn0suc 16734. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nn0suc0	|- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )

Distinct variable group:   x, A

Proof of Theorem bj-nn0suc0

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2239	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
2		eqeq1 2239	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
3	2	rexeqbi1dv 2754	. . 3 |- ( y = A -> ( E. x e. y y = suc x <-> E. x e. A A = suc x ) )
4	1, 3	orbi12d 801	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. y y = suc x ) <-> ( A = (/) \/ E. x e. A A = suc x ) ) )
5		tru 1402	. . 3 |- T.
6		trud 1414	. . . 4 |- ( T. -> T. )
7	6	rgenw 2597	. . 3 |- A. z e. om ( T. -> T. )
8		bdeq0 16637	. . . . 5 |- BOUNDED y = (/)
9		bdeqsuc 16651	. . . . . 6 |- BOUNDED y = suc x
10	9	ax-bdex 16589	. . . . 5 |- BOUNDED E. x e. y y = suc x
11	8, 10	ax-bdor 16586	. . . 4 |- BOUNDED ( y = (/) \/ E. x e. y y = suc x )
12		nfv 1577	. . . 4 |- F/ y T.
13		orc 720	. . . . 5 |- ( y = (/) -> ( y = (/) \/ E. x e. y y = suc x ) )
14	13	a1d 22	. . . 4 |- ( y = (/) -> ( T. -> ( y = (/) \/ E. x e. y y = suc x ) ) )
15		trud 1414	. . . . 5 |- ( -. ( y = z -> -. ( y = (/) \/ E. x e. y y = suc x ) ) -> T. )
16	15	expi 643	. . . 4 |- ( y = z -> ( ( y = (/) \/ E. x e. y y = suc x ) -> T. ) )
17		vex 2816	. . . . . . . . 9 |- z e. _V
18	17	sucid 4538	. . . . . . . 8 |- z e. suc z
19		eleq2 2296	. . . . . . . 8 |- ( y = suc z -> ( z e. y <-> z e. suc z ) )
20	18, 19	mpbiri 168	. . . . . . 7 |- ( y = suc z -> z e. y )
21		suceq 4523	. . . . . . . . 9 |- ( x = z -> suc x = suc z )
22	21	eqeq2d 2244	. . . . . . . 8 |- ( x = z -> ( y = suc x <-> y = suc z ) )
23	22	rspcev 2921	. . . . . . 7 |- ( ( z e. y /\ y = suc z ) -> E. x e. y y = suc x )
24	20, 23	mpancom 422	. . . . . 6 |- ( y = suc z -> E. x e. y y = suc x )
25	24	olcd 742	. . . . 5 |- ( y = suc z -> ( y = (/) \/ E. x e. y y = suc x ) )
26	25	a1d 22	. . . 4 |- ( y = suc z -> ( T. -> ( y = (/) \/ E. x e. y y = suc x ) ) )
27	11, 12, 12, 12, 14, 16, 26	bj-bdfindis 16717	. . 3 |- ( ( T. /\ A. z e. om ( T. -> T. ) ) -> A. y e. om ( y = (/) \/ E. x e. y y = suc x ) )
28	5, 7, 27	mp2an 426	. 2 |- A. y e. om ( y = (/) \/ E. x e. y y = suc x )
29	4, 28	vtoclri 2892	1 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 716   = wceq 1398   T. wtru 1399   e. wcel 2203   A.wral 2520   E.wrex 2521   (/)c0 3508   suc csuc 4486   omcom 4712

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 2205  ax-14 2206  ax-ext 2214  ax-nul 4236  ax-pr 4322  ax-un 4554  ax-bd0 16583  ax-bdim 16584  ax-bdan 16585  ax-bdor 16586  ax-bdn 16587  ax-bdal 16588  ax-bdex 16589  ax-bdeq 16590  ax-bdel 16591  ax-bdsb 16592  ax-bdsep 16654  ax-infvn 16711

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-sn 3695  df-pr 3696  df-uni 3915  df-int 3950  df-suc 4492  df-iom 4713  df-bdc 16611  df-bj-ind 16697

This theorem is referenced by:  bj-nn0suc  16734