Theorem el1o 4230

Description: Membership in ordinal one.

Assertion

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

Proof of Theorem el1o

Step	Hyp	Ref	Expression
1		df1o2 4224	. . 3 |- 1o = {(/)}
2	1	eleq2i 1575	. 2 |- (A e. 1o <-> A e. {(/)})
3		0ex 2762	. . 3 |- (/) e. V
4	3	elsnc2 2482	. 2 |- (A e. {(/)} <-> A = (/))
5	2, 4	bitri 171	1 |- (A e. 1o <-> A = (/))

Syntax hints:   <-> wb 144   = wceq 988   e. wcel 990  (/)c0 2324  {csn 2454  1oc1o 4212

This theorem is referenced by:  0lt1o 4231  oelim2 4306  map1 4517  cfsuc 5004

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-nul 2761

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-v 1850  df-dif 2093  df-un 2094  df-nul 2325  df-sn 2457  df-pr 2458  df-suc 3009  df-1o 4217

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