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Theorem nsuceq0g 4104
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 3205 . . 3 ¬ A
2 sucidg 4102 . . . 4 (A 𝑉A suc A)
3 eleq2 2083 . . . 4 (suc A = ∅ → (A suc AA ∅))
42, 3syl5ibcom 144 . . 3 (A 𝑉 → (suc A = ∅ → A ∅))
51, 4mtoi 577 . 2 (A 𝑉 → ¬ suc A = ∅)
65neneqad 2262 1 (A 𝑉 → suc A ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228   wcel 1374  wne 2186  c0 3201  suc csuc 4051
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-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  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-v 2537  df-dif 2897  df-un 2899  df-nul 3202  df-sn 3356  df-suc 4057
This theorem is referenced by:  onsucelsucexmid  4199  peano3  4246  frec0g  5902  2on0  5925
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