Theorem bj-nn0suc0 16271

Description: Constructive proof of a variant of nn0suc 4695. For a constructive proof of nn0suc 4695, see bj-nn0suc 16285. (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 2236	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
2		eqeq1 2236	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
3	2	rexeqbi1dv 2741	. . 3 |- ( y = A -> ( E. x e. y y = suc x <-> E. x e. A A = suc x ) )
4	1, 3	orbi12d 798	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. y y = suc x ) <-> ( A = (/) \/ E. x e. A A = suc x ) ) )
5		tru 1399	. . 3 |- T.
6		trud 1411	. . . 4 |- ( T. -> T. )
7	6	rgenw 2585	. . 3 |- A. z e. om ( T. -> T. )
8		bdeq0 16188	. . . . 5 |- BOUNDED y = (/)
9		bdeqsuc 16202	. . . . . 6 |- BOUNDED y = suc x
10	9	ax-bdex 16140	. . . . 5 |- BOUNDED E. x e. y y = suc x
11	8, 10	ax-bdor 16137	. . . 4 |- BOUNDED ( y = (/) \/ E. x e. y y = suc x )
12		nfv 1574	. . . 4 |- F/ y T.
13		orc 717	. . . . 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 1411	. . . . 5 |- ( -. ( y = z -> -. ( y = (/) \/ E. x e. y y = suc x ) ) -> T. )
16	15	expi 641	. . . 4 |- ( y = z -> ( ( y = (/) \/ E. x e. y y = suc x ) -> T. ) )
17		vex 2802	. . . . . . . . 9 |- z e. _V
18	17	sucid 4507	. . . . . . . 8 |- z e. suc z
19		eleq2 2293	. . . . . . . 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 4492	. . . . . . . . 9 |- ( x = z -> suc x = suc z )
22	21	eqeq2d 2241	. . . . . . . 8 |- ( x = z -> ( y = suc x <-> y = suc z ) )
23	22	rspcev 2907	. . . . . . 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 739	. . . . 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 16268	. . 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 2878	1 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 713   = wceq 1395   T. wtru 1396   e. wcel 2200   A.wral 2508   E.wrex 2509   (/)c0 3491   suc csuc 4455   omcom 4681

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-nul 4209  ax-pr 4292  ax-un 4523  ax-bd0 16134  ax-bdim 16135  ax-bdan 16136  ax-bdor 16137  ax-bdn 16138  ax-bdal 16139  ax-bdex 16140  ax-bdeq 16141  ax-bdel 16142  ax-bdsb 16143  ax-bdsep 16205  ax-infvn 16262

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-pr 3673  df-uni 3888  df-int 3923  df-suc 4461  df-iom 4682  df-bdc 16162  df-bj-ind 16248

This theorem is referenced by:  bj-nn0suc  16285