Theorem bj-nn0suc0 14973

Description: Constructive proof of a variant of nn0suc 4615. For a constructive proof of nn0suc 4615, see bj-nn0suc 14987. (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 2194	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
2		eqeq1 2194	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
3	2	rexeqbi1dv 2692	. . 3 |- ( y = A -> ( E. x e. y y = suc x <-> E. x e. A A = suc x ) )
4	1, 3	orbi12d 794	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. y y = suc x ) <-> ( A = (/) \/ E. x e. A A = suc x ) ) )
5		tru 1367	. . 3 |- T.
6		trud 1379	. . . 4 |- ( T. -> T. )
7	6	rgenw 2542	. . 3 |- A. z e. om ( T. -> T. )
8		bdeq0 14890	. . . . 5 |- BOUNDED y = (/)
9		bdeqsuc 14904	. . . . . 6 |- BOUNDED y = suc x
10	9	ax-bdex 14842	. . . . 5 |- BOUNDED E. x e. y y = suc x
11	8, 10	ax-bdor 14839	. . . 4 |- BOUNDED ( y = (/) \/ E. x e. y y = suc x )
12		nfv 1538	. . . 4 |- F/ y T.
13		orc 713	. . . . 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 1379	. . . . 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 2752	. . . . . . . . 9 |- z e. _V
18	17	sucid 4429	. . . . . . . 8 |- z e. suc z
19		eleq2 2251	. . . . . . . 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 4414	. . . . . . . . 9 |- ( x = z -> suc x = suc z )
22	21	eqeq2d 2199	. . . . . . . 8 |- ( x = z -> ( y = suc x <-> y = suc z ) )
23	22	rspcev 2853	. . . . . . 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 735	. . . . 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 14970	. . 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 2824	1 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 709   = wceq 1363   T. wtru 1364   e. wcel 2158   A.wral 2465   E.wrex 2466   (/)c0 3434   suc csuc 4377   omcom 4601

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-nul 4141  ax-pr 4221  ax-un 4445  ax-bd0 14836  ax-bdim 14837  ax-bdan 14838  ax-bdor 14839  ax-bdn 14840  ax-bdal 14841  ax-bdex 14842  ax-bdeq 14843  ax-bdel 14844  ax-bdsb 14845  ax-bdsep 14907  ax-infvn 14964

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-sn 3610  df-pr 3611  df-uni 3822  df-int 3857  df-suc 4383  df-iom 4602  df-bdc 14864  df-bj-ind 14950

This theorem is referenced by:  bj-nn0suc  14987