Theorem bj-nn0suc0 12733

Description: Constructive proof of a variant of nn0suc 4456. For a constructive proof of nn0suc 4456, see bj-nn0suc 12747. (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 2106	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
2		eqeq1 2106	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
3	2	rexeqbi1dv 2593	. . 3 |- ( y = A -> ( E. x e. y y = suc x <-> E. x e. A A = suc x ) )
4	1, 3	orbi12d 748	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. y y = suc x ) <-> ( A = (/) \/ E. x e. A A = suc x ) ) )
5		tru 1303	. . 3 |- T.
6		a1tru 1315	. . . 4 |- ( T. -> T. )
7	6	rgenw 2446	. . 3 |- A. z e. om ( T. -> T. )
8		bdeq0 12646	. . . . 5 |- BOUNDED y = (/)
9		bdeqsuc 12660	. . . . . 6 |- BOUNDED y = suc x
10	9	ax-bdex 12598	. . . . 5 |- BOUNDED E. x e. y y = suc x
11	8, 10	ax-bdor 12595	. . . 4 |- BOUNDED ( y = (/) \/ E. x e. y y = suc x )
12		nfv 1476	. . . 4 |- F/ y T.
13		orc 674	. . . . 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		a1tru 1315	. . . . 5 |- ( -. ( y = z -> -. ( y = (/) \/ E. x e. y y = suc x ) ) -> T. )
16	15	expi 607	. . . 4 |- ( y = z -> ( ( y = (/) \/ E. x e. y y = suc x ) -> T. ) )
17		vex 2644	. . . . . . . . 9 |- z e. _V
18	17	sucid 4277	. . . . . . . 8 |- z e. suc z
19		eleq2 2163	. . . . . . . 8 |- ( y = suc z -> ( z e. y <-> z e. suc z ) )
20	18, 19	mpbiri 167	. . . . . . 7 |- ( y = suc z -> z e. y )
21		suceq 4262	. . . . . . . . 9 |- ( x = z -> suc x = suc z )
22	21	eqeq2d 2111	. . . . . . . 8 |- ( x = z -> ( y = suc x <-> y = suc z ) )
23	22	rspcev 2744	. . . . . . 7 |- ( ( z e. y /\ y = suc z ) -> E. x e. y y = suc x )
24	20, 23	mpancom 416	. . . . . 6 |- ( y = suc z -> E. x e. y y = suc x )
25	24	olcd 694	. . . . 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 12730	. . 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 420	. 2 |- A. y e. om ( y = (/) \/ E. x e. y y = suc x )
29	4, 28	vtoclri 2716	1 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 670   = wceq 1299   T. wtru 1300   e. wcel 1448   A.wral 2375   E.wrex 2376   (/)c0 3310   suc csuc 4225   omcom 4442

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-nul 3994  ax-pr 4069  ax-un 4293  ax-bd0 12592  ax-bdim 12593  ax-bdan 12594  ax-bdor 12595  ax-bdn 12596  ax-bdal 12597  ax-bdex 12598  ax-bdeq 12599  ax-bdel 12600  ax-bdsb 12601  ax-bdsep 12663  ax-infvn 12724

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-sn 3480  df-pr 3481  df-uni 3684  df-int 3719  df-suc 4231  df-iom 4443  df-bdc 12620  df-bj-ind 12710

This theorem is referenced by:  bj-nn0suc  12747