Theorem nn0suc 4518

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 2146	. . 3 |- ( y = (/) -> ( y = (/) <-> (/) = (/) ) )
2		eqeq1 2146	. . . 4 |- ( y = (/) -> ( y = suc x <-> (/) = suc x ) )
3	2	rexbidv 2438	. . 3 |- ( y = (/) -> ( E. x e. om y = suc x <-> E. x e. om (/) = suc x ) )
4	1, 3	orbi12d 782	. 2 |- ( y = (/) -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( (/) = (/) \/ E. x e. om (/) = suc x ) ) )
5		eqeq1 2146	. . 3 |- ( y = z -> ( y = (/) <-> z = (/) ) )
6		eqeq1 2146	. . . 4 |- ( y = z -> ( y = suc x <-> z = suc x ) )
7	6	rexbidv 2438	. . 3 |- ( y = z -> ( E. x e. om y = suc x <-> E. x e. om z = suc x ) )
8	5, 7	orbi12d 782	. 2 |- ( y = z -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( z = (/) \/ E. x e. om z = suc x ) ) )
9		eqeq1 2146	. . 3 |- ( y = suc z -> ( y = (/) <-> suc z = (/) ) )
10		eqeq1 2146	. . . 4 |- ( y = suc z -> ( y = suc x <-> suc z = suc x ) )
11	10	rexbidv 2438	. . 3 |- ( y = suc z -> ( E. x e. om y = suc x <-> E. x e. om suc z = suc x ) )
12	9, 11	orbi12d 782	. 2 |- ( y = suc z -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( suc z = (/) \/ E. x e. om suc z = suc x ) ) )
13		eqeq1 2146	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
14		eqeq1 2146	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
15	14	rexbidv 2438	. . 3 |- ( y = A -> ( E. x e. om y = suc x <-> E. x e. om A = suc x ) )
16	13, 15	orbi12d 782	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( A = (/) \/ E. x e. om A = suc x ) ) )
17		eqid 2139	. . 3 |- (/) = (/)
18	17	orci 720	. 2 |- ( (/) = (/) \/ E. x e. om (/) = suc x )
19		eqid 2139	. . . . 5 |- suc z = suc z
20		suceq 4324	. . . . . . 7 |- ( x = z -> suc x = suc z )
21	20	eqeq2d 2151	. . . . . 6 |- ( x = z -> ( suc z = suc x <-> suc z = suc z ) )
22	21	rspcev 2789	. . . . 5 |- ( ( z e. om /\ suc z = suc z ) -> E. x e. om suc z = suc x )
23	19, 22	mpan2 421	. . . 4 |- ( z e. om -> E. x e. om suc z = suc x )
24	23	olcd 723	. . 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 4514	1 |- ( A e. om -> ( A = (/) \/ E. x e. om A = suc x ) )

Syntax hints:   -> wi 4   \/ wo 697   = wceq 1331   e. wcel 1480   E.wrex 2417   (/)c0 3363   suc csuc 4287   omcom 4504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505

This theorem is referenced by:  nnsuc  4529  nnpredcl  4536  frecabcl  6296  nnsucuniel  6391  nneneq  6751  phpm  6759  dif1enen  6774  fin0  6779  fin0or  6780  diffisn  6787