Theorem 1ne0 4132

Description: Ordinal one is not equal to ordinal zero.

Assertion

Ref	Expression
1ne0	|- 1o =/= (/)

Proof of Theorem 1ne0

Step	Hyp	Ref	Expression
1		0ex 2706	. . 3 |- (/) e. V
2	1	snnz 2454	. 2 |- {(/)} =/= (/)
3		df1o2 4130	. . 3 |- 1o = {(/)}
4	3	neeq1i 1589	. 2 |- (1o =/= (/) <-> {(/)} =/= (/))
5	2, 4	mpbir 190	1 |- 1o =/= (/)

Syntax hints:   =/= wne 1582  (/)c0 2276  {csn 2405  1oc1o 4118

This theorem is referenced by:  xp01disj 4133  card1 4813  unxpdom2 4825  sucxpdom 4826  cdacomen 4909  1pi 4991

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-nul 2705

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-nul 2277  df-sn 2408  df-pr 2409  df-suc 2949  df-1o 4123

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