Theorem bj-nn0suc0 16649

Description: Constructive proof of a variant of nn0suc 4708. For a constructive proof of nn0suc 4708, see bj-nn0suc 16663. (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 2238	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
2		eqeq1 2238	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
3	2	rexeqbi1dv 2744	. . 3 |- ( y = A -> ( E. x e. y y = suc x <-> E. x e. A A = suc x ) )
4	1, 3	orbi12d 801	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. y y = suc x ) <-> ( A = (/) \/ E. x e. A A = suc x ) ) )
5		tru 1402	. . 3 |- T.
6		trud 1414	. . . 4 |- ( T. -> T. )
7	6	rgenw 2588	. . 3 |- A. z e. om ( T. -> T. )
8		bdeq0 16566	. . . . 5 |- BOUNDED y = (/)
9		bdeqsuc 16580	. . . . . 6 |- BOUNDED y = suc x
10	9	ax-bdex 16518	. . . . 5 |- BOUNDED E. x e. y y = suc x
11	8, 10	ax-bdor 16515	. . . 4 |- BOUNDED ( y = (/) \/ E. x e. y y = suc x )
12		nfv 1577	. . . 4 |- F/ y T.
13		orc 720	. . . . 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 1414	. . . . 5 |- ( -. ( y = z -> -. ( y = (/) \/ E. x e. y y = suc x ) ) -> T. )
16	15	expi 643	. . . 4 |- ( y = z -> ( ( y = (/) \/ E. x e. y y = suc x ) -> T. ) )
17		vex 2806	. . . . . . . . 9 |- z e. _V
18	17	sucid 4520	. . . . . . . 8 |- z e. suc z
19		eleq2 2295	. . . . . . . 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 4505	. . . . . . . . 9 |- ( x = z -> suc x = suc z )
22	21	eqeq2d 2243	. . . . . . . 8 |- ( x = z -> ( y = suc x <-> y = suc z ) )
23	22	rspcev 2911	. . . . . . 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 742	. . . . 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 16646	. . 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 2882	1 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 716   = wceq 1398   T. wtru 1399   e. wcel 2202   A.wral 2511   E.wrex 2512   (/)c0 3496   suc csuc 4468   omcom 4694

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-nul 4220  ax-pr 4305  ax-un 4536  ax-bd0 16512  ax-bdim 16513  ax-bdan 16514  ax-bdor 16515  ax-bdn 16516  ax-bdal 16517  ax-bdex 16518  ax-bdeq 16519  ax-bdel 16520  ax-bdsb 16521  ax-bdsep 16583  ax-infvn 16640

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-sn 3679  df-pr 3680  df-uni 3899  df-int 3934  df-suc 4474  df-iom 4695  df-bdc 16540  df-bj-ind 16626

This theorem is referenced by:  bj-nn0suc  16663