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Theorem nnsuc 4281
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 2203 . 2 (A ≠ ∅ ↔ ¬ A = ∅)
2 nn0suc 4270 . . . 4 (A 𝜔 → (A = ∅ x 𝜔 A = suc x))
32ord 642 . . 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 1242   wcel 1390  wne 2201  wrex 2301  c0 3218  suc csuc 4068  𝜔com 4256
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257
This theorem is referenced by: (None)
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