Theorem bj-nn0suc0 14705

Description: Constructive proof of a variant of nn0suc 4604. For a constructive proof of nn0suc 4604, see bj-nn0suc 14719. (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 2184	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
2		eqeq1 2184	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
3	2	rexeqbi1dv 2682	. . 3 |- ( y = A -> ( E. x e. y y = suc x <-> E. x e. A A = suc x ) )
4	1, 3	orbi12d 793	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. y y = suc x ) <-> ( A = (/) \/ E. x e. A A = suc x ) ) )
5		tru 1357	. . 3 |- T.
6		a1tru 1369	. . . 4 |- ( T. -> T. )
7	6	rgenw 2532	. . 3 |- A. z e. om ( T. -> T. )
8		bdeq0 14622	. . . . 5 |- BOUNDED y = (/)
9		bdeqsuc 14636	. . . . . 6 |- BOUNDED y = suc x
10	9	ax-bdex 14574	. . . . 5 |- BOUNDED E. x e. y y = suc x
11	8, 10	ax-bdor 14571	. . . 4 |- BOUNDED ( y = (/) \/ E. x e. y y = suc x )
12		nfv 1528	. . . 4 |- F/ y T.
13		orc 712	. . . . 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 1369	. . . . 5 |- ( -. ( y = z -> -. ( y = (/) \/ E. x e. y y = suc x ) ) -> T. )
16	15	expi 638	. . . 4 |- ( y = z -> ( ( y = (/) \/ E. x e. y y = suc x ) -> T. ) )
17		vex 2741	. . . . . . . . 9 |- z e. _V
18	17	sucid 4418	. . . . . . . 8 |- z e. suc z
19		eleq2 2241	. . . . . . . 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 4403	. . . . . . . . 9 |- ( x = z -> suc x = suc z )
22	21	eqeq2d 2189	. . . . . . . 8 |- ( x = z -> ( y = suc x <-> y = suc z ) )
23	22	rspcev 2842	. . . . . . 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 734	. . . . 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 14702	. . 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 2813	1 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 708   = wceq 1353   T. wtru 1354   e. wcel 2148   A.wral 2455   E.wrex 2456   (/)c0 3423   suc csuc 4366   omcom 4590

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4130  ax-pr 4210  ax-un 4434  ax-bd0 14568  ax-bdim 14569  ax-bdan 14570  ax-bdor 14571  ax-bdn 14572  ax-bdal 14573  ax-bdex 14574  ax-bdeq 14575  ax-bdel 14576  ax-bdsb 14577  ax-bdsep 14639  ax-infvn 14696

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-sn 3599  df-pr 3600  df-uni 3811  df-int 3846  df-suc 4372  df-iom 4591  df-bdc 14596  df-bj-ind 14682

This theorem is referenced by:  bj-nn0suc  14719