Theorem nn0suc 4604

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 2184	. . 3 |- ( y = (/) -> ( y = (/) <-> (/) = (/) ) )
2		eqeq1 2184	. . . 4 |- ( y = (/) -> ( y = suc x <-> (/) = suc x ) )
3	2	rexbidv 2478	. . 3 |- ( y = (/) -> ( E. x e. om y = suc x <-> E. x e. om (/) = suc x ) )
4	1, 3	orbi12d 793	. 2 |- ( y = (/) -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( (/) = (/) \/ E. x e. om (/) = suc x ) ) )
5		eqeq1 2184	. . 3 |- ( y = z -> ( y = (/) <-> z = (/) ) )
6		eqeq1 2184	. . . 4 |- ( y = z -> ( y = suc x <-> z = suc x ) )
7	6	rexbidv 2478	. . 3 |- ( y = z -> ( E. x e. om y = suc x <-> E. x e. om z = suc x ) )
8	5, 7	orbi12d 793	. 2 |- ( y = z -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( z = (/) \/ E. x e. om z = suc x ) ) )
9		eqeq1 2184	. . 3 |- ( y = suc z -> ( y = (/) <-> suc z = (/) ) )
10		eqeq1 2184	. . . 4 |- ( y = suc z -> ( y = suc x <-> suc z = suc x ) )
11	10	rexbidv 2478	. . 3 |- ( y = suc z -> ( E. x e. om y = suc x <-> E. x e. om suc z = suc x ) )
12	9, 11	orbi12d 793	. 2 |- ( y = suc z -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( suc z = (/) \/ E. x e. om suc z = suc x ) ) )
13		eqeq1 2184	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
14		eqeq1 2184	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
15	14	rexbidv 2478	. . 3 |- ( y = A -> ( E. x e. om y = suc x <-> E. x e. om A = suc x ) )
16	13, 15	orbi12d 793	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( A = (/) \/ E. x e. om A = suc x ) ) )
17		eqid 2177	. . 3 |- (/) = (/)
18	17	orci 731	. 2 |- ( (/) = (/) \/ E. x e. om (/) = suc x )
19		eqid 2177	. . . . 5 |- suc z = suc z
20		suceq 4403	. . . . . . 7 |- ( x = z -> suc x = suc z )
21	20	eqeq2d 2189	. . . . . 6 |- ( x = z -> ( suc z = suc x <-> suc z = suc z ) )
22	21	rspcev 2842	. . . . 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 734	. . 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 4600	1 |- ( A e. om -> ( A = (/) \/ E. x e. om A = suc x ) )

Syntax hints:   -> wi 4   \/ wo 708   = wceq 1353   e. wcel 2148   E.wrex 2456   (/)c0 3423   suc csuc 4366   omcom 4590

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-uni 3811  df-int 3846  df-suc 4372  df-iom 4591

This theorem is referenced by:  nnsuc  4616  nnpredcl  4623  frecabcl  6400  nnsucuniel  6496  nneneq  6857  phpm  6865  dif1enen  6880  fin0  6885  fin0or  6886  diffisn  6893