Theorem bj-nn0suc0 13985

Description: Constructive proof of a variant of nn0suc 4588. For a constructive proof of nn0suc 4588, see bj-nn0suc 13999. (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 2177	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
2		eqeq1 2177	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
3	2	rexeqbi1dv 2674	. . 3 |- ( y = A -> ( E. x e. y y = suc x <-> E. x e. A A = suc x ) )
4	1, 3	orbi12d 788	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. y y = suc x ) <-> ( A = (/) \/ E. x e. A A = suc x ) ) )
5		tru 1352	. . 3 |- T.
6		a1tru 1364	. . . 4 |- ( T. -> T. )
7	6	rgenw 2525	. . 3 |- A. z e. om ( T. -> T. )
8		bdeq0 13902	. . . . 5 |- BOUNDED y = (/)
9		bdeqsuc 13916	. . . . . 6 |- BOUNDED y = suc x
10	9	ax-bdex 13854	. . . . 5 |- BOUNDED E. x e. y y = suc x
11	8, 10	ax-bdor 13851	. . . 4 |- BOUNDED ( y = (/) \/ E. x e. y y = suc x )
12		nfv 1521	. . . 4 |- F/ y T.
13		orc 707	. . . . 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		a1tru 1364	. . . . 5 |- ( -. ( y = z -> -. ( y = (/) \/ E. x e. y y = suc x ) ) -> T. )
16	15	expi 633	. . . 4 |- ( y = z -> ( ( y = (/) \/ E. x e. y y = suc x ) -> T. ) )
17		vex 2733	. . . . . . . . 9 |- z e. _V
18	17	sucid 4402	. . . . . . . 8 |- z e. suc z
19		eleq2 2234	. . . . . . . 8 |- ( y = suc z -> ( z e. y <-> z e. suc z ) )
20	18, 19	mpbiri 167	. . . . . . 7 |- ( y = suc z -> z e. y )
21		suceq 4387	. . . . . . . . 9 |- ( x = z -> suc x = suc z )
22	21	eqeq2d 2182	. . . . . . . 8 |- ( x = z -> ( y = suc x <-> y = suc z ) )
23	22	rspcev 2834	. . . . . . 7 |- ( ( z e. y /\ y = suc z ) -> E. x e. y y = suc x )
24	20, 23	mpancom 420	. . . . . 6 |- ( y = suc z -> E. x e. y y = suc x )
25	24	olcd 729	. . . . 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 13982	. . 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 424	. 2 |- A. y e. om ( y = (/) \/ E. x e. y y = suc x )
29	4, 28	vtoclri 2805	1 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 703   = wceq 1348   T. wtru 1349   e. wcel 2141   A.wral 2448   E.wrex 2449   (/)c0 3414   suc csuc 4350   omcom 4574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-nul 4115  ax-pr 4194  ax-un 4418  ax-bd0 13848  ax-bdim 13849  ax-bdan 13850  ax-bdor 13851  ax-bdn 13852  ax-bdal 13853  ax-bdex 13854  ax-bdeq 13855  ax-bdel 13856  ax-bdsb 13857  ax-bdsep 13919  ax-infvn 13976

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-suc 4356  df-iom 4575  df-bdc 13876  df-bj-ind 13962

This theorem is referenced by:  bj-nn0suc  13999