Theorem nnsuc 2389

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

Assertion

Ref	Expression
nnsuc	⊢ ((A ∈ ω ∧ ¬ A = ∅) → ∃x ∈ ω A = suc x)

Distinct variable group(s):   x,A

Proof of Theorem nnsuc

Step	Hyp	Ref	Expression
1		nnlim 2385	. . . 4 ⊢ (A ∈ ω → ¬ Lim A)
2	1	adantr 306	. . 3 ⊢ ((A ∈ ω ∧ ¬ A = ∅) → ¬ Lim A)
3		orduninsuc 2365	. . . . . 6 ⊢ (Ord A → (A = ∪A ↔ ¬ ∃x ∈ On A = suc x))
4	3	adantr 306	. . . . 5 ⊢ ((Ord A ∧ ¬ A = ∅) → (A = ∪A ↔ ¬ ∃x ∈ On A = suc x))
5		df-lim 2204	. . . . . . . 8 ⊢ (Lim A ↔ (Ord A ∧ ¬ A = ∅ ∧ A = ∪A))
6	5	biimpr 134	. . . . . . 7 ⊢ ((Ord A ∧ ¬ A = ∅ ∧ A = ∪A) → Lim A)
7	6	3exp 611	. . . . . 6 ⊢ (Ord A → (¬ A = ∅ → (A = ∪A → Lim A)))
8	7	imp 277	. . . . 5 ⊢ ((Ord A ∧ ¬ A = ∅) → (A = ∪A → Lim A))
9	4, 8	sylbird 180	. . . 4 ⊢ ((Ord A ∧ ¬ A = ∅) → (¬ ∃x ∈ On A = suc x → Lim A))
10		nnord 2381	. . . 4 ⊢ (A ∈ ω → Ord A)
11	9, 10	sylan 343	. . 3 ⊢ ((A ∈ ω ∧ ¬ A = ∅) → (¬ ∃x ∈ On A = suc x → Lim A))
12	2, 11	mt3d 101	. 2 ⊢ ((A ∈ ω ∧ ¬ A = ∅) → ∃x ∈ On A = suc x)
13		eleq1 1149	. . . . . . . . 9 ⊢ (A = suc x → (A ∈ ω ↔ suc x ∈ ω))
14	13	biimpcd 137	. . . . . . . 8 ⊢ (A ∈ ω → (A = suc x → suc x ∈ ω))
15		peano2b 2388	. . . . . . . 8 ⊢ (x ∈ ω ↔ suc x ∈ ω)
16	14, 15	syl6ibr 186	. . . . . . 7 ⊢ (A ∈ ω → (A = suc x → x ∈ ω))
17	16	ancrd 247	. . . . . 6 ⊢ (A ∈ ω → (A = suc x → (x ∈ ω ∧ A = suc x)))
18	17	adantld 307	. . . . 5 ⊢ (A ∈ ω → ((x ∈ On ∧ A = suc x) → (x ∈ ω ∧ A = suc x)))
19	18	19.22dv 947	. . . 4 ⊢ (A ∈ ω → (∃x(x ∈ On ∧ A = suc x) → ∃x(x ∈ ω ∧ A = suc x)))
20		df-rex 1206	. . . 4 ⊢ (∃x ∈ On A = suc x ↔ ∃x(x ∈ On ∧ A = suc x))
21		df-rex 1206	. . . 4 ⊢ (∃x ∈ ω A = suc x ↔ ∃x(x ∈ ω ∧ A = suc x))
22	19, 20, 21	3imtr4g 426	. . 3 ⊢ (A ∈ ω → (∃x ∈ On A = suc x → ∃x ∈ ω A = suc x))
23	22	adantr 306	. 2 ⊢ ((A ∈ ω ∧ ¬ A = ∅) → (∃x ∈ On A = suc x → ∃x ∈ ω A = suc x))
24	12, 23	mpd 46	1 ⊢ ((A ∈ ω ∧ ¬ A = ∅) → ∃x ∈ ω A = suc x)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   ∧ w3a 581  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ∅c0 1707  ∪cuni 1919  Ord word 2198  Oncon0 2199  Lim wlim 2200  suc csuc 2201  ωcom 2372

This theorem is referenced by:  peano5 2394  nn0suc 2395  inf3lemd 3463

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373

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