Theorem nnsuc 3143

Description: A non-zero natural number is a successor.

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 3139	. . . 4 |- (A e. om -> -. Lim A)
2	1	adantr 389	. . 3 |- ((A e. om /\ A =/= (/)) -> -. Lim A)
3		orduninsuc 3109	. . . . . 6 |- (Ord A -> (A = U.A <-> -. E.x e. On A = suc x))
4	3	adantr 389	. . . . 5 |- ((Ord A /\ A =/= (/)) -> (A = U.A <-> -. E.x e. On A = suc x))
5		df-lim 2948	. . . . . . 7 |- (Lim A <-> (Ord A /\ A =/= (/) /\ A = U.A))
6	5	biimpr 152	. . . . . 6 |- ((Ord A /\ A =/= (/) /\ A = U.A) -> Lim A)
7	6	3expia 834	. . . . 5 |- ((Ord A /\ A =/= (/)) -> (A = U.A -> Lim A))
8	4, 7	sylbird 205	. . . 4 |- ((Ord A /\ A =/= (/)) -> (-. E.x e. On A = suc x -> Lim A))
9		nnord 3135	. . . 4 |- (A e. om -> Ord A)
10	8, 9	sylan 448	. . 3 |- ((A e. om /\ A =/= (/)) -> (-. E.x e. On A = suc x -> Lim A))
11	2, 10	mt3d 114	. 2 |- ((A e. om /\ A =/= (/)) -> E.x e. On A = suc x)
12		eleq1 1531	. . . . . . . . 9 |- (A = suc x -> (A e. om <-> suc x e. om))
13	12	biimpcd 155	. . . . . . . 8 |- (A e. om -> (A = suc x -> suc x e. om))
14		peano2b 3142	. . . . . . . 8 |- (x e. om <-> suc x e. om)
15	13, 14	syl6ibr 213	. . . . . . 7 |- (A e. om -> (A = suc x -> x e. om))
16	15	ancrd 299	. . . . . 6 |- (A e. om -> (A = suc x -> (x e. om /\ A = suc x)))
17	16	adantld 390	. . . . 5 |- (A e. om -> ((x e. On /\ A = suc x) -> (x e. om /\ A = suc x)))
18	17	19.22dv 1288	. . . 4 |- (A e. om -> (E.x(x e. On /\ A = suc x) -> E.x(x e. om /\ A = suc x)))
19		df-rex 1647	. . . 4 |- (E.x e. On A = suc x <-> E.x(x e. On /\ A = suc x))
20		df-rex 1647	. . . 4 |- (E.x e. om A = suc x <-> E.x(x e. om /\ A = suc x))
21	18, 19, 20	3imtr4g 552	. . 3 |- (A e. om -> (E.x e. On A = suc x -> E.x e. om A = suc x))
22	21	adantr 389	. 2 |- ((A e. om /\ A =/= (/)) -> (E.x e. On A = suc x -> E.x e. om A = suc x))
23	11, 22	mpd 26	1 |- ((A e. om /\ A =/= (/)) -> E.x e. om A = suc x)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  E.wex 978   =/= wne 1582  E.wrex 1643  (/)c0 2276  U.cuni 2498  Ord word 2942  Oncon0 2943  Lim wlim 2944  suc csuc 2945  omcom 3126

This theorem is referenced by:  peano5 3148  nn0suc 3149  inf3lemd 4592

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127

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