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Theorem nsuceq0g 4120
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 3222 . . 3 ¬ A
2 sucidg 4118 . . . 4 (A 𝑉A suc A)
3 eleq2 2098 . . . 4 (suc A = ∅ → (A suc AA ∅))
42, 3syl5ibcom 144 . . 3 (A 𝑉 → (suc A = ∅ → A ∅))
51, 4mtoi 589 . 2 (A 𝑉 → ¬ suc A = ∅)
65neneqad 2278 1 (A 𝑉 → suc A ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  wne 2201  c0 3218  suc csuc 4067
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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  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-v 2553  df-dif 2914  df-un 2916  df-nul 3219  df-sn 3372  df-suc 4073
This theorem is referenced by:  onsucelsucexmid  4214  peano3  4261  frec0g  5916  2on0  5942
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