Theorem 0cnsuc 4401

Description: Cardinal zero is not a successor. Compare Theorem X.1.2 of [Rosser] p. 275. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
0cnsuc	|- (A +o 1c) =/= 0c

Proof of Theorem 0cnsuc

Step	Hyp	Ref	Expression
1		0nelsuc 4400	. . 3 |- -. (/) e. (A +o 1c)
2		0ex 4110	. . . . . 6 |- (/) e. _V
3	2	snid 3760	. . . . 5 |- (/) e. {(/)}
4		df-0c 4377	. . . . 5 |- 0c = {(/)}
5	3, 4	eleqtrri 2426	. . . 4 |- (/) e. 0c
6		eleq2 2414	. . . 4 |- ((A +o 1c) = 0c -> ((/) e. (A +o 1c) <-> (/) e. 0c))
7	5, 6	mpbiri 224	. . 3 |- ((A +o 1c) = 0c -> (/) e. (A +o 1c))
8	1, 7	mto 167	. 2 |- -. (A +o 1c) = 0c
9		df-ne 2518	. 2 |- ((A +o 1c) =/= 0c <-> -. (A +o 1c) = 0c)
10	8, 9	mpbir 200	1 |- (A +o 1c) =/= 0c

Syntax hints:  -. wn 3   = wceq 1642   e. wcel 1710   =/= wne 2516  (/)c0 3550  {csn 3737  1_cc1c 4134  0_cc0c 4374   +o 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:  peano3  4404  ltfinirr  4457  evenodddisj  4516  0cnelphi  4597  addceq0  6219  1ne0c  6241  2ne0c  6242  nnltp1c  6262  nnc3n3p1  6278  nchoicelem12  6300  nchoicelem14  6302  nchoicelem17  6305  fnfreclem2  6318  fnfreclem3  6319