Theorem 1ne0 4200

Description: Ordinal one is not equal to ordinal zero.

Assertion

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

Proof of Theorem 1ne0

Step	Hyp	Ref	Expression
1		0ex 2766	. . 3 |- (/) e. V
2	1	snnz 2512	. 2 |- {(/)} =/= (/)
3		df1o2 4198	. . 3 |- 1o = {(/)}
4	3	neeq1i 1639	. 2 |- (1o =/= (/) <-> {(/)} =/= (/))
5	2, 4	mpbir 197	1 |- 1o =/= (/)

Syntax hints:   =/= wne 1632  (/)c0 2331  {csn 2461  1oc1o 4186

This theorem is referenced by:  xp01disj 4201  card1 4896  unxpdom2 4910  sucxpdom 4911  cdacomen 4994  1pi 5076

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-nul 2765

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-v 1859  df-dif 2100  df-un 2101  df-nul 2332  df-sn 2464  df-pr 2465  df-suc 3011  df-1o 4191

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