Theorem bj-nn0suc0 13137

Description: Constructive proof of a variant of nn0suc 4513. For a constructive proof of nn0suc 4513, see bj-nn0suc 13151. (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 2144	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
2		eqeq1 2144	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
3	2	rexeqbi1dv 2633	. . 3 |- ( y = A -> ( E. x e. y y = suc x <-> E. x e. A A = suc x ) )
4	1, 3	orbi12d 782	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. y y = suc x ) <-> ( A = (/) \/ E. x e. A A = suc x ) ) )
5		tru 1335	. . 3 |- T.
6		a1tru 1347	. . . 4 |- ( T. -> T. )
7	6	rgenw 2485	. . 3 |- A. z e. om ( T. -> T. )
8		bdeq0 13054	. . . . 5 |- BOUNDED y = (/)
9		bdeqsuc 13068	. . . . . 6 |- BOUNDED y = suc x
10	9	ax-bdex 13006	. . . . 5 |- BOUNDED E. x e. y y = suc x
11	8, 10	ax-bdor 13003	. . . 4 |- BOUNDED ( y = (/) \/ E. x e. y y = suc x )
12		nfv 1508	. . . 4 |- F/ y T.
13		orc 701	. . . . 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 1347	. . . . 5 |- ( -. ( y = z -> -. ( y = (/) \/ E. x e. y y = suc x ) ) -> T. )
16	15	expi 627	. . . 4 |- ( y = z -> ( ( y = (/) \/ E. x e. y y = suc x ) -> T. ) )
17		vex 2684	. . . . . . . . 9 |- z e. _V
18	17	sucid 4334	. . . . . . . 8 |- z e. suc z
19		eleq2 2201	. . . . . . . 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 4319	. . . . . . . . 9 |- ( x = z -> suc x = suc z )
22	21	eqeq2d 2149	. . . . . . . 8 |- ( x = z -> ( y = suc x <-> y = suc z ) )
23	22	rspcev 2784	. . . . . . 7 |- ( ( z e. y /\ y = suc z ) -> E. x e. y y = suc x )
24	20, 23	mpancom 418	. . . . . 6 |- ( y = suc z -> E. x e. y y = suc x )
25	24	olcd 723	. . . . 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 13134	. . 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 422	. 2 |- A. y e. om ( y = (/) \/ E. x e. y y = suc x )
29	4, 28	vtoclri 2756	1 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 697   = wceq 1331   T. wtru 1332   e. wcel 1480   A.wral 2414   E.wrex 2415   (/)c0 3358   suc csuc 4282   omcom 4499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-nul 4049  ax-pr 4126  ax-un 4350  ax-bd0 13000  ax-bdim 13001  ax-bdan 13002  ax-bdor 13003  ax-bdn 13004  ax-bdal 13005  ax-bdex 13006  ax-bdeq 13007  ax-bdel 13008  ax-bdsb 13009  ax-bdsep 13071  ax-infvn 13128

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-sn 3528  df-pr 3529  df-uni 3732  df-int 3767  df-suc 4288  df-iom 4500  df-bdc 13028  df-bj-ind 13114

This theorem is referenced by:  bj-nn0suc  13151