Theorem nn0suc 4615

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 2194	. . 3 |- ( y = (/) -> ( y = (/) <-> (/) = (/) ) )
2		eqeq1 2194	. . . 4 |- ( y = (/) -> ( y = suc x <-> (/) = suc x ) )
3	2	rexbidv 2488	. . 3 |- ( y = (/) -> ( E. x e. om y = suc x <-> E. x e. om (/) = suc x ) )
4	1, 3	orbi12d 794	. 2 |- ( y = (/) -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( (/) = (/) \/ E. x e. om (/) = suc x ) ) )
5		eqeq1 2194	. . 3 |- ( y = z -> ( y = (/) <-> z = (/) ) )
6		eqeq1 2194	. . . 4 |- ( y = z -> ( y = suc x <-> z = suc x ) )
7	6	rexbidv 2488	. . 3 |- ( y = z -> ( E. x e. om y = suc x <-> E. x e. om z = suc x ) )
8	5, 7	orbi12d 794	. 2 |- ( y = z -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( z = (/) \/ E. x e. om z = suc x ) ) )
9		eqeq1 2194	. . 3 |- ( y = suc z -> ( y = (/) <-> suc z = (/) ) )
10		eqeq1 2194	. . . 4 |- ( y = suc z -> ( y = suc x <-> suc z = suc x ) )
11	10	rexbidv 2488	. . 3 |- ( y = suc z -> ( E. x e. om y = suc x <-> E. x e. om suc z = suc x ) )
12	9, 11	orbi12d 794	. 2 |- ( y = suc z -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( suc z = (/) \/ E. x e. om suc z = suc x ) ) )
13		eqeq1 2194	. . 3 |- ( y = A -> ( y = (/) <-> A = (/) ) )
14		eqeq1 2194	. . . 4 |- ( y = A -> ( y = suc x <-> A = suc x ) )
15	14	rexbidv 2488	. . 3 |- ( y = A -> ( E. x e. om y = suc x <-> E. x e. om A = suc x ) )
16	13, 15	orbi12d 794	. 2 |- ( y = A -> ( ( y = (/) \/ E. x e. om y = suc x ) <-> ( A = (/) \/ E. x e. om A = suc x ) ) )
17		eqid 2187	. . 3 |- (/) = (/)
18	17	orci 732	. 2 |- ( (/) = (/) \/ E. x e. om (/) = suc x )
19		eqid 2187	. . . . 5 |- suc z = suc z
20		suceq 4414	. . . . . . 7 |- ( x = z -> suc x = suc z )
21	20	eqeq2d 2199	. . . . . 6 |- ( x = z -> ( suc z = suc x <-> suc z = suc z ) )
22	21	rspcev 2853	. . . . 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 735	. . 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 4611	1 |- ( A e. om -> ( A = (/) \/ E. x e. om A = suc x ) )

Syntax hints:   -> wi 4   \/ wo 709   = wceq 1363   e. wcel 2158   E.wrex 2466   (/)c0 3434   suc csuc 4377   omcom 4601

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-uni 3822  df-int 3857  df-suc 4383  df-iom 4602

This theorem is referenced by:  nnsuc  4627  nnpredcl  4634  frecabcl  6414  nnsucuniel  6510  nneneq  6871  phpm  6879  dif1enen  6894  fin0  6899  fin0or  6900  diffisn  6907