Theorem peano3 4405

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

Step	Hyp	Ref	Expression
1		0cnsuc 4402	. 2 ⊢ (A +c 1c) ≠ 0c
2	1	a1i 10	1 ⊢ (A ∈ Nn → (A +c 1c) ≠ 0c)

Syntax hints:   → wi 4   ∈ wcel 1710   ≠ wne 2517  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376

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 4079  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-1c 4137  df-0c 4378  df-addc 4379

This theorem is referenced by: (None)