Theorem nn0suc 4731

Description: A natural number is either 0 or a successor. Similar theorems for arbitrary sets or real numbers will not be provable (without the law of the excluded middle), but equality of natural numbers is decidable. (Contributed by NM, 27-May-1998.)

Assertion

Ref	Expression
nn0suc	|- ( A e. om -> ( A = (/) \/ E. x e. om A = suc x ) )

Distinct variable group:   x, A

Proof of Theorem nn0suc

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2241	. . 3 |- ( y = (/) -> ( y = (/) <-> (/) = (/) ) )
2		eqeq1 2241	. . . 4 |- ( y = (/) -> ( y = suc x <-> (/) = suc x ) )
3	2	rexbidv 2545	. . 3 |- ( y = (/) -> ( E. x e. om y = suc x <-> E. x e. om (/) = suc x ) )
4	1, 3	orbi12d 801	. 2 |- ( y = (/) -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( (/) = (/) \/ E. x e. om (/) = suc x ) ) )
5		eqeq1 2241	. . 3 |- ( y = z -> ( y = (/) <-> z = (/) ) )
6		eqeq1 2241	. . . 4 |- ( y = z -> ( y = suc x <-> z = suc x ) )
7	6	rexbidv 2545	. . 3 |- ( y = z -> ( E. x e. om y = suc x <-> E. x e. om z = suc x ) )
8	5, 7	orbi12d 801	. 2 |- ( y = z -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( z = (/) \/ E. x e. om z = suc x ) ) )
9		eqeq1 2241	. . 3 |- ( y = suc z -> ( y = (/) <-> suc z = (/) ) )
10		eqeq1 2241	. . . 4 |- ( y = suc z -> ( y = suc x <-> suc z = suc x ) )
11	10	rexbidv 2545	. . 3 |- ( y = suc z -> ( E. x e. om y = suc x <-> E. x e. om suc z = suc x ) )
12	9, 11	orbi12d 801	. 2 |- ( y = suc z -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( suc z = (/) \/ E. x e. om suc z = suc x ) ) )
13		eqeq1 2241	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
14		eqeq1 2241	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
15	14	rexbidv 2545	. . 3 |- ( y = A -> ( E. x e. om y = suc x <-> E. x e. om A = suc x ) )
16	13, 15	orbi12d 801	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( A = (/) \/ E. x e. om A = suc x ) ) )
17		eqid 2234	. . 3 |- (/) = (/)
18	17	orci 739	. 2 |- ( (/) = (/) \/ E. x e. om (/) = suc x )
19		eqid 2234	. . . . 5 |- suc z = suc z
20		suceq 4528	. . . . . . 7 |- ( x = z -> suc x = suc z )
21	20	eqeq2d 2246	. . . . . 6 |- ( x = z -> ( suc z = suc x <-> suc z = suc z ) )
22	21	rspcev 2923	. . . . 5 |- ( ( z e. om /\ suc z = suc z ) -> E. x e. om suc z = suc x )
23	19, 22	mpan2 425	. . . 4 |- ( z e. om -> E. x e. om suc z = suc x )
24	23	olcd 742	. . 3 |- ( z e. om -> ( suc z = (/) \/ E. x e. om suc z = suc x ) )
25	24	a1d 22	. 2 |- ( z e. om -> ( ( z = (/) \/ E. x e. om z = suc x ) -> ( suc z = (/) \/ E. x e. om suc z = suc x ) ) )
26	4, 8, 12, 16, 18, 25	finds 4727	1 |- ( A e. om -> ( A = (/) \/ E. x e. om A = suc x ) )

Syntax hints:   -> wi 4   \/ wo 716   = wceq 1398   e. wcel 2205   E.wrex 2523   (/)c0 3512   suc csuc 4491   omcom 4717

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-int 3955  df-suc 4497  df-iom 4718

This theorem is referenced by:  nnsuc  4743  nnpredcl  4750  frecabcl  6643  nnsucuniel  6741  nneneq  7124  phpm  7133  dif1enen  7150  fin0  7155  fin0or  7156  diffisn  7163