Theorem nn0suc 4670

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 2214	. . 3 |- ( y = (/) -> ( y = (/) <-> (/) = (/) ) )
2		eqeq1 2214	. . . 4 |- ( y = (/) -> ( y = suc x <-> (/) = suc x ) )
3	2	rexbidv 2509	. . 3 |- ( y = (/) -> ( E. x e. om y = suc x <-> E. x e. om (/) = suc x ) )
4	1, 3	orbi12d 795	. 2 |- ( y = (/) -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( (/) = (/) \/ E. x e. om (/) = suc x ) ) )
5		eqeq1 2214	. . 3 |- ( y = z -> ( y = (/) <-> z = (/) ) )
6		eqeq1 2214	. . . 4 |- ( y = z -> ( y = suc x <-> z = suc x ) )
7	6	rexbidv 2509	. . 3 |- ( y = z -> ( E. x e. om y = suc x <-> E. x e. om z = suc x ) )
8	5, 7	orbi12d 795	. 2 |- ( y = z -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( z = (/) \/ E. x e. om z = suc x ) ) )
9		eqeq1 2214	. . 3 |- ( y = suc z -> ( y = (/) <-> suc z = (/) ) )
10		eqeq1 2214	. . . 4 |- ( y = suc z -> ( y = suc x <-> suc z = suc x ) )
11	10	rexbidv 2509	. . 3 |- ( y = suc z -> ( E. x e. om y = suc x <-> E. x e. om suc z = suc x ) )
12	9, 11	orbi12d 795	. 2 |- ( y = suc z -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( suc z = (/) \/ E. x e. om suc z = suc x ) ) )
13		eqeq1 2214	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
14		eqeq1 2214	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
15	14	rexbidv 2509	. . 3 |- ( y = A -> ( E. x e. om y = suc x <-> E. x e. om A = suc x ) )
16	13, 15	orbi12d 795	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( A = (/) \/ E. x e. om A = suc x ) ) )
17		eqid 2207	. . 3 |- (/) = (/)
18	17	orci 733	. 2 |- ( (/) = (/) \/ E. x e. om (/) = suc x )
19		eqid 2207	. . . . 5 |- suc z = suc z
20		suceq 4467	. . . . . . 7 |- ( x = z -> suc x = suc z )
21	20	eqeq2d 2219	. . . . . 6 |- ( x = z -> ( suc z = suc x <-> suc z = suc z ) )
22	21	rspcev 2884	. . . . 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 736	. . 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 4666	1 |- ( A e. om -> ( A = (/) \/ E. x e. om A = suc x ) )

Syntax hints:   -> wi 4   \/ wo 710   = wceq 1373   e. wcel 2178   E.wrex 2487   (/)c0 3468   suc csuc 4430   omcom 4656

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-int 3900  df-suc 4436  df-iom 4657

This theorem is referenced by:  nnsuc  4682  nnpredcl  4689  frecabcl  6508  nnsucuniel  6604  nneneq  6979  phpm  6988  dif1enen  7003  fin0  7008  fin0or  7009  diffisn  7016