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Theorem nnsuc 4265
Description: A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
Assertion
Ref Expression
nnsuc ((A 𝜔 A ≠ ∅) → x 𝜔 A = suc x)
Distinct variable group:   x,A

Proof of Theorem nnsuc
StepHypRef Expression
1 df-ne 2188 . 2 (A ≠ ∅ ↔ ¬ A = ∅)
2 nn0suc 4254 . . . 4 (A 𝜔 → (A = ∅ x 𝜔 A = suc x))
32ord 630 . . 3 (A 𝜔 → (¬ A = ∅ → x 𝜔 A = suc x))
43imp 115 . 2 ((A 𝜔 ¬ A = ∅) → x 𝜔 A = suc x)
51, 4sylan2b 271 1 ((A 𝜔 A ≠ ∅) → x 𝜔 A = suc x)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1228   wcel 1374  wne 2186  wrex 2285  c0 3201  suc csuc 4051  𝜔com 4240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241
This theorem is referenced by: (None)
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