Theorem bj-nn0suc0 15512

Description: Constructive proof of a variant of nn0suc 4637. For a constructive proof of nn0suc 4637, see bj-nn0suc 15526. (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 2200	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
2		eqeq1 2200	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
3	2	rexeqbi1dv 2703	. . 3 |- ( y = A -> ( E. x e. y y = suc x <-> E. x e. A A = suc x ) )
4	1, 3	orbi12d 794	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. y y = suc x ) <-> ( A = (/) \/ E. x e. A A = suc x ) ) )
5		tru 1368	. . 3 |- T.
6		trud 1380	. . . 4 |- ( T. -> T. )
7	6	rgenw 2549	. . 3 |- A. z e. om ( T. -> T. )
8		bdeq0 15429	. . . . 5 |- BOUNDED y = (/)
9		bdeqsuc 15443	. . . . . 6 |- BOUNDED y = suc x
10	9	ax-bdex 15381	. . . . 5 |- BOUNDED E. x e. y y = suc x
11	8, 10	ax-bdor 15378	. . . 4 |- BOUNDED ( y = (/) \/ E. x e. y y = suc x )
12		nfv 1539	. . . 4 |- F/ y T.
13		orc 713	. . . . 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 1380	. . . . 5 |- ( -. ( y = z -> -. ( y = (/) \/ E. x e. y y = suc x ) ) -> T. )
16	15	expi 639	. . . 4 |- ( y = z -> ( ( y = (/) \/ E. x e. y y = suc x ) -> T. ) )
17		vex 2763	. . . . . . . . 9 |- z e. _V
18	17	sucid 4449	. . . . . . . 8 |- z e. suc z
19		eleq2 2257	. . . . . . . 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 4434	. . . . . . . . 9 |- ( x = z -> suc x = suc z )
22	21	eqeq2d 2205	. . . . . . . 8 |- ( x = z -> ( y = suc x <-> y = suc z ) )
23	22	rspcev 2865	. . . . . . 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 735	. . . . 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 15509	. . 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 2836	1 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 709   = wceq 1364   T. wtru 1365   e. wcel 2164   A.wral 2472   E.wrex 2473   (/)c0 3447   suc csuc 4397   omcom 4623

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-nul 4156  ax-pr 4239  ax-un 4465  ax-bd0 15375  ax-bdim 15376  ax-bdan 15377  ax-bdor 15378  ax-bdn 15379  ax-bdal 15380  ax-bdex 15381  ax-bdeq 15382  ax-bdel 15383  ax-bdsb 15384  ax-bdsep 15446  ax-infvn 15503

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-sn 3625  df-pr 3626  df-uni 3837  df-int 3872  df-suc 4403  df-iom 4624  df-bdc 15403  df-bj-ind 15489

This theorem is referenced by:  bj-nn0suc  15526