Theorem 1ne0 4142

Description: Ordinal one is not equal to ordinal zero.

Assertion

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

Proof of Theorem 1ne0

Step	Hyp	Ref	Expression
1		0ex 2711	. . 3 |- (/) e. V
2	1	snnz 2458	. 2 |- {(/)} =/= (/)
3		df1o2 4140	. . 3 |- 1o = {(/)}
4	3	neeq1i 1592	. 2 |- (1o =/= (/) <-> {(/)} =/= (/))
5	2, 4	mpbir 190	1 |- 1o =/= (/)

Syntax hints:   =/= wne 1585  (/)c0 2280  {csn 2409  1oc1o 4128

This theorem is referenced by:  xp01disj 4143  card1 4833  unxpdom2 4845  sucxpdom 4846  cdacomen 4929  1pi 5011

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-nul 2710

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-nul 2281  df-sn 2412  df-pr 2413  df-suc 2954  df-1o 4133

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