Theorem bj-nn0suc0 16890

Description: Constructive proof of a variant of nn0suc 4746. For a constructive proof of nn0suc 4746, see bj-nn0suc 16904. (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 2245	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
2		eqeq1 2245	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
3	2	rexeqbi1dv 2762	. . 3 |- ( y = A -> ( E. x e. y y = suc x <-> E. x e. A A = suc x ) )
4	1, 3	orbi12d 805	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. y y = suc x ) <-> ( A = (/) \/ E. x e. A A = suc x ) ) )
5		tru 1406	. . 3 |- T.
6		trud 1418	. . . 4 |- ( T. -> T. )
7	6	rgenw 2605	. . 3 |- A. z e. om ( T. -> T. )
8		bdeq0 16807	. . . . 5 |- BOUNDED y = (/)
9		bdeqsuc 16821	. . . . . 6 |- BOUNDED y = suc x
10	9	ax-bdex 16759	. . . . 5 |- BOUNDED E. x e. y y = suc x
11	8, 10	ax-bdor 16756	. . . 4 |- BOUNDED ( y = (/) \/ E. x e. y y = suc x )
12		nfv 1581	. . . 4 |- F/ y T.
13		orc 724	. . . . 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 1418	. . . . 5 |- ( -. ( y = z -> -. ( y = (/) \/ E. x e. y y = suc x ) ) -> T. )
16	15	expi 647	. . . 4 |- ( y = z -> ( ( y = (/) \/ E. x e. y y = suc x ) -> T. ) )
17		vex 2824	. . . . . . . . 9 |- z e. _V
18	17	sucid 4557	. . . . . . . 8 |- z e. suc z
19		eleq2 2302	. . . . . . . 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 4542	. . . . . . . . 9 |- ( x = z -> suc x = suc z )
22	21	eqeq2d 2250	. . . . . . . 8 |- ( x = z -> ( y = suc x <-> y = suc z ) )
23	22	rspcev 2929	. . . . . . 7 |- ( ( z e. y /\ y = suc z ) -> E. x e. y y = suc x )
24	20, 23	mpancom 426	. . . . . 6 |- ( y = suc z -> E. x e. y y = suc x )
25	24	olcd 746	. . . . 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 16887	. . 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 430	. 2 |- A. y e. om ( y = (/) \/ E. x e. y y = suc x )
29	4, 28	vtoclri 2900	1 |- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 720   = wceq 1402   T. wtru 1403   e. wcel 2209   A.wral 2528   E.wrex 2529   (/)c0 3520   suc csuc 4505   omcom 4732

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-nul 4254  ax-pr 4341  ax-un 4573  ax-bd0 16753  ax-bdim 16754  ax-bdan 16755  ax-bdor 16756  ax-bdn 16757  ax-bdal 16758  ax-bdex 16759  ax-bdeq 16760  ax-bdel 16761  ax-bdsb 16762  ax-bdsep 16824  ax-infvn 16881

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-suc 4511  df-iom 4733  df-bdc 16781  df-bj-ind 16867

This theorem is referenced by:  bj-nn0suc  16904