Theorem bj-nn0suc0 15886

Description: Constructive proof of a variant of nn0suc 4652. For a constructive proof of nn0suc 4652, see bj-nn0suc 15900. (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 2212	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
2		eqeq1 2212	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
3	2	rexeqbi1dv 2715	. . 3 |- ( y = A -> ( E. x e. y y = suc x <-> E. x e. A A = suc x ) )
4	1, 3	orbi12d 795	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. y y = suc x ) <-> ( A = (/) \/ E. x e. A A = suc x ) ) )
5		tru 1377	. . 3 |- T.
6		trud 1389	. . . 4 |- ( T. -> T. )
7	6	rgenw 2561	. . 3 |- A. z e. om ( T. -> T. )
8		bdeq0 15803	. . . . 5 |- BOUNDED y = (/)
9		bdeqsuc 15817	. . . . . 6 |- BOUNDED y = suc x
10	9	ax-bdex 15755	. . . . 5 |- BOUNDED E. x e. y y = suc x
11	8, 10	ax-bdor 15752	. . . 4 |- BOUNDED ( y = (/) \/ E. x e. y y = suc x )
12		nfv 1551	. . . 4 |- F/ y T.
13		orc 714	. . . . 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 1389	. . . . 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 2775	. . . . . . . . 9 |- z e. _V
18	17	sucid 4464	. . . . . . . 8 |- z e. suc z
19		eleq2 2269	. . . . . . . 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 4449	. . . . . . . . 9 |- ( x = z -> suc x = suc z )
22	21	eqeq2d 2217	. . . . . . . 8 |- ( x = z -> ( y = suc x <-> y = suc z ) )
23	22	rspcev 2877	. . . . . . 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 736	. . . . 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 15883	. . 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 2848	1 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 710   = wceq 1373   T. wtru 1374   e. wcel 2176   A.wral 2484   E.wrex 2485   (/)c0 3460   suc csuc 4412   omcom 4638

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-nul 4170  ax-pr 4253  ax-un 4480  ax-bd0 15749  ax-bdim 15750  ax-bdan 15751  ax-bdor 15752  ax-bdn 15753  ax-bdal 15754  ax-bdex 15755  ax-bdeq 15756  ax-bdel 15757  ax-bdsb 15758  ax-bdsep 15820  ax-infvn 15877

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-suc 4418  df-iom 4639  df-bdc 15777  df-bj-ind 15863

This theorem is referenced by:  bj-nn0suc  15900