Theorem nnsuc 7082

Description: A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)

Assertion

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

Distinct variable group:   x, A

Proof of Theorem nnsuc

Step	Hyp	Ref	Expression
1		nnlim 7078	. . . 4 |- ( A e. om -> -. Lim A )
2	1	adantr 481	. . 3 |- ( ( A e. om /\ A =/= (/) ) -> -. Lim A )
3		nnord 7073	. . . 4 |- ( A e. om -> Ord A )
4		orduninsuc 7043	. . . . . 6 |- ( Ord A -> ( A = U. A <-> -. E. x e. On A = suc x ) )
5	4	adantr 481	. . . . 5 |- ( ( Ord A /\ A =/= (/) ) -> ( A = U. A <-> -. E. x e. On A = suc x ) )
6		df-lim 5728	. . . . . . 7 |- ( Lim A <-> ( Ord A /\ A =/= (/) /\ A = U. A ) )
7	6	biimpri 218	. . . . . 6 |- ( ( Ord A /\ A =/= (/) /\ A = U. A ) -> Lim A )
8	7	3expia 1267	. . . . 5 |- ( ( Ord A /\ A =/= (/) ) -> ( A = U. A -> Lim A ) )
9	5, 8	sylbird 250	. . . 4 |- ( ( Ord A /\ A =/= (/) ) -> ( -. E. x e. On A = suc x -> Lim A ) )
10	3, 9	sylan 488	. . 3 |- ( ( A e. om /\ A =/= (/) ) -> ( -. E. x e. On A = suc x -> Lim A ) )
11	2, 10	mt3d 140	. 2 |- ( ( A e. om /\ A =/= (/) ) -> E. x e. On A = suc x )
12		eleq1 2689	. . . . . . . 8 |- ( A = suc x -> ( A e. om <-> suc x e. om ) )
13	12	biimpcd 239	. . . . . . 7 |- ( A e. om -> ( A = suc x -> suc x e. om ) )
14		peano2b 7081	. . . . . . 7 |- ( x e. om <-> suc x e. om )
15	13, 14	syl6ibr 242	. . . . . 6 |- ( A e. om -> ( A = suc x -> x e. om ) )
16	15	ancrd 577	. . . . 5 |- ( A e. om -> ( A = suc x -> ( x e. om /\ A = suc x ) ) )
17	16	adantld 483	. . . 4 |- ( A e. om -> ( ( x e. On /\ A = suc x ) -> ( x e. om /\ A = suc x ) ) )
18	17	reximdv2 3014	. . 3 |- ( A e. om -> ( E. x e. On A = suc x -> E. x e. om A = suc x ) )
19	18	adantr 481	. 2 |- ( ( A e. om /\ A =/= (/) ) -> ( E. x e. On A = suc x -> E. x e. om A = suc x ) )
20	11, 19	mpd 15	1 |- ( ( A e. om /\ A =/= (/) ) -> E. x e. om A = suc x )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   (/)c0 3915   U.cuni 4436   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725   omcom 7065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-om 7066

This theorem is referenced by:  peano5  7089  nn0suc  7090  inf3lemd  8524  infpssrlem4  9128  fin1a2lem6  9227  bnj158  30797  bnj1098  30854  bnj594  30982