Theorem bj-nn0suc0 15848

Description: Constructive proof of a variant of nn0suc 4651. For a constructive proof of nn0suc 4651, see bj-nn0suc 15862. (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 2211	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
2		eqeq1 2211	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
3	2	rexeqbi1dv 2714	. . 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 1376	. . 3 |- T.
6		trud 1388	. . . 4 |- ( T. -> T. )
7	6	rgenw 2560	. . 3 |- A. z e. om ( T. -> T. )
8		bdeq0 15765	. . . . 5 |- BOUNDED y = (/)
9		bdeqsuc 15779	. . . . . 6 |- BOUNDED y = suc x
10	9	ax-bdex 15717	. . . . 5 |- BOUNDED E. x e. y y = suc x
11	8, 10	ax-bdor 15714	. . . 4 |- BOUNDED ( y = (/) \/ E. x e. y y = suc x )
12		nfv 1550	. . . 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 1388	. . . . 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 2774	. . . . . . . . 9 |- z e. _V
18	17	sucid 4463	. . . . . . . 8 |- z e. suc z
19		eleq2 2268	. . . . . . . 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 4448	. . . . . . . . 9 |- ( x = z -> suc x = suc z )
22	21	eqeq2d 2216	. . . . . . . 8 |- ( x = z -> ( y = suc x <-> y = suc z ) )
23	22	rspcev 2876	. . . . . . 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 15845	. . 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 2847	1 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 709   = wceq 1372   T. wtru 1373   e. wcel 2175   A.wral 2483   E.wrex 2484   (/)c0 3459   suc csuc 4411   omcom 4637

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-nul 4169  ax-pr 4252  ax-un 4479  ax-bd0 15711  ax-bdim 15712  ax-bdan 15713  ax-bdor 15714  ax-bdn 15715  ax-bdal 15716  ax-bdex 15717  ax-bdeq 15718  ax-bdel 15719  ax-bdsb 15720  ax-bdsep 15782  ax-infvn 15839

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-sn 3638  df-pr 3639  df-uni 3850  df-int 3885  df-suc 4417  df-iom 4638  df-bdc 15739  df-bj-ind 15825

This theorem is referenced by:  bj-nn0suc  15862