Theorem el1o 4136

Description: Membership in ordinal one.

Assertion

Ref	Expression
el1o	|- (A e. 1o <-> A = (/))

Proof of Theorem el1o

Step	Hyp	Ref	Expression
1		df1o2 4130	. . 3 |- 1o = {(/)}
2	1	eleq2i 1535	. 2 |- (A e. 1o <-> A e. {(/)})
3		0ex 2706	. . 3 |- (/) e. V
4	3	elsnc2 2433	. 2 |- (A e. {(/)} <-> A = (/))
5	2, 4	bitr 173	1 |- (A e. 1o <-> A = (/))

Syntax hints:   <-> wb 146   = wceq 954   e. wcel 956  (/)c0 2276  {csn 2405  1oc1o 4118

This theorem is referenced by:  0lt1o 4137  oelim2 4212  map1 4417  cfsuc 4895

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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