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Theorem nsuceq0g 4075
Description: No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.)
Assertion
Ref Expression
nsuceq0g (A 𝑉 → suc A ≠ ∅)

Proof of Theorem nsuceq0g
StepHypRef Expression
1 noel 3202 . . 3 ¬ A
2 sucidg 4073 . . . 4 (A 𝑉A suc A)
3 eleq2 2082 . . . 4 (suc A = ∅ → (A suc AA ∅))
42, 3syl5ibcom 144 . . 3 (A 𝑉 → (suc A = ∅ → A ∅))
51, 4mtoi 577 . 2 (A 𝑉 → ¬ suc A = ∅)
65neneqad 2259 1 (A 𝑉 → suc A ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1230   wcel 1376  wne 2185  c0 3198  suc csuc 4023
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 1318  ax-7 1319  ax-gen 1320  ax-ie1 1365  ax-ie2 1366  ax-8 1378  ax-10 1379  ax-11 1380  ax-i12 1381  ax-bnd 1382  ax-4 1383  ax-17 1402  ax-i9 1406  ax-ial 1411  ax-i5r 1412  ax-ext 2003
This theorem depends on definitions:  df-bi 110  df-tru 1233  df-nf 1332  df-sb 1629  df-clab 2008  df-cleq 2014  df-clel 2017  df-nfc 2148  df-ne 2187  df-v 2534  df-dif 2894  df-un 2896  df-nul 3199  df-sn 3329  df-suc 4028
This theorem is referenced by:  onsucelsucexmid  4170  peano3  4217  frec0g  5870  2on0  5893
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