Theorem bj-nn0suc0 16055

Description: Constructive proof of a variant of nn0suc 4665. For a constructive proof of nn0suc 4665, see bj-nn0suc 16069. (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 2213	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
2		eqeq1 2213	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
3	2	rexeqbi1dv 2716	. . 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 2562	. . 3 |- A. z e. om ( T. -> T. )
8		bdeq0 15972	. . . . 5 |- BOUNDED y = (/)
9		bdeqsuc 15986	. . . . . 6 |- BOUNDED y = suc x
10	9	ax-bdex 15924	. . . . 5 |- BOUNDED E. x e. y y = suc x
11	8, 10	ax-bdor 15921	. . . 4 |- BOUNDED ( y = (/) \/ E. x e. y y = suc x )
12		nfv 1552	. . . 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 2776	. . . . . . . . 9 |- z e. _V
18	17	sucid 4477	. . . . . . . 8 |- z e. suc z
19		eleq2 2270	. . . . . . . 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 4462	. . . . . . . . 9 |- ( x = z -> suc x = suc z )
22	21	eqeq2d 2218	. . . . . . . 8 |- ( x = z -> ( y = suc x <-> y = suc z ) )
23	22	rspcev 2881	. . . . . . 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 16052	. . 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 2852	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 2177   A.wral 2485   E.wrex 2486   (/)c0 3464   suc csuc 4425   omcom 4651

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-nul 4181  ax-pr 4264  ax-un 4493  ax-bd0 15918  ax-bdim 15919  ax-bdan 15920  ax-bdor 15921  ax-bdn 15922  ax-bdal 15923  ax-bdex 15924  ax-bdeq 15925  ax-bdel 15926  ax-bdsb 15927  ax-bdsep 15989  ax-infvn 16046

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-sn 3644  df-pr 3645  df-uni 3860  df-int 3895  df-suc 4431  df-iom 4652  df-bdc 15946  df-bj-ind 16032

This theorem is referenced by:  bj-nn0suc  16069