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Theorem peano3 4404
 Description: The successor of a finite cardinal is not zero. (Contributed by SF, 14-Jan-2015.)
Assertion
Ref Expression
peano3 (A Nn → (A +c 1c) ≠ 0c)

Proof of Theorem peano3
StepHypRef Expression
1 0cnsuc 4401 . 2 (A +c 1c) ≠ 0c
21a1i 10 1 (A Nn → (A +c 1c) ≠ 0c)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710   ≠ wne 2516  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   +c cplc 4375 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-1c 4136  df-0c 4377  df-addc 4378 This theorem is referenced by: (None)
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