Theorem bj-nn0suc0 10745

Description: Constructive proof of a variant of nn0suc 4345. For a constructive proof of nn0suc 4345, see bj-nn0suc 10759. (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 2087	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
2		eqeq1 2087	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
3	2	rexeqbi1dv 2558	. . 3 |- ( y = A -> ( E. x e. y y = suc x <-> E. x e. A A = suc x ) )
4	1, 3	orbi12d 739	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. y y = suc x ) <-> ( A = (/) \/ E. x e. A A = suc x ) ) )
5		tru 1288	. . 3 |- T.
6		a1tru 1300	. . . 4 |- ( T. -> T. )
7	6	rgenw 2418	. . 3 |- A. z e. om ( T. -> T. )
8		bdeq0 10658	. . . . 5 |- BOUNDED y = (/)
9		bdeqsuc 10672	. . . . . 6 |- BOUNDED y = suc x
10	9	ax-bdex 10610	. . . . 5 |- BOUNDED E. x e. y y = suc x
11	8, 10	ax-bdor 10607	. . . 4 |- BOUNDED ( y = (/) \/ E. x e. y y = suc x )
12		nfv 1461	. . . 4 |- F/ y T.
13		orc 665	. . . . 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 1300	. . . . 5 |- ( -. ( y = z -> -. ( y = (/) \/ E. x e. y y = suc x ) ) -> T. )
16	15	expi 599	. . . 4 |- ( y = z -> ( ( y = (/) \/ E. x e. y y = suc x ) -> T. ) )
17		vex 2604	. . . . . . . . 9 |- z e. _V
18	17	sucid 4172	. . . . . . . 8 |- z e. suc z
19		eleq2 2142	. . . . . . . 8 |- ( y = suc z -> ( z e. y <-> z e. suc z ) )
20	18, 19	mpbiri 166	. . . . . . 7 |- ( y = suc z -> z e. y )
21		suceq 4157	. . . . . . . . 9 |- ( x = z -> suc x = suc z )
22	21	eqeq2d 2092	. . . . . . . 8 |- ( x = z -> ( y = suc x <-> y = suc z ) )
23	22	rspcev 2701	. . . . . . 7 |- ( ( z e. y /\ y = suc z ) -> E. x e. y y = suc x )
24	20, 23	mpancom 413	. . . . . 6 |- ( y = suc z -> E. x e. y y = suc x )
25	24	olcd 685	. . . . 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 10742	. . 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 416	. 2 |- A. y e. om ( y = (/) \/ E. x e. y y = suc x )
29	4, 28	vtoclri 2673	1 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 661   = wceq 1284   T. wtru 1285   e. wcel 1433   A.wral 2348   E.wrex 2349   (/)c0 3251   suc csuc 4120   omcom 4331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-nul 3904  ax-pr 3964  ax-un 4188  ax-bd0 10604  ax-bdim 10605  ax-bdan 10606  ax-bdor 10607  ax-bdn 10608  ax-bdal 10609  ax-bdex 10610  ax-bdeq 10611  ax-bdel 10612  ax-bdsb 10613  ax-bdsep 10675  ax-infvn 10736

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-suc 4126  df-iom 4332  df-bdc 10632  df-bj-ind 10722

This theorem is referenced by:  bj-nn0suc  10759