Theorem 1ne0c 6242

Description: Cardinal one is not zero. (Contributed by SF, 4-Mar-2015.)

Assertion

Ref	Expression
1ne0c	|- 1c =/= 0c

Proof of Theorem 1ne0c

Step	Hyp	Ref	Expression
1		addcid2 4408	. 2 |- (0c +o 1c) = 1c
2		0cnsuc 4402	. 2 |- (0c +o 1c) =/= 0c
3	1, 2	eqnetrri 2536	1 |- 1c =/= 0c

Syntax hints:   =/= wne 2517  1_cc1c 4135  0_cc0c 4375   +o 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-3an 936  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-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-pr 3743  df-opk 4059  df-1c 4137  df-pw1 4138  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-sik 4193  df-ssetk 4194  df-0c 4378  df-addc 4379

This theorem is referenced by:  nnc3n3p1  6279