Theorem el1o 6546

Description: Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)

Assertion

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

Proof of Theorem el1o

Step	Hyp	Ref	Expression
1		df1o2 6538	. . 3 |- 1o = { (/) }
2	1	eleq2i 2274	. 2 |- ( A e. 1o <-> A e. { (/) } )
3		0ex 4187	. . 3 |- (/) e. _V
4	3	elsn2 3677	. 2 |- ( A e. { (/) } <-> A = (/) )
5	2, 4	bitri 184	1 |- ( A e. 1o <-> A = (/) )

Syntax hints:   <-> wb 105   = wceq 1373   e. wcel 2178   (/)c0 3468   {csn 3643   1oc1o 6518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-nul 4186

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-un 3178  df-nul 3469  df-sn 3649  df-suc 4436  df-1o 6525

This theorem is referenced by:  0lt1o  6549  map0e  6796  map1  6928  omp1eomlem  7222  ctmlemr  7236  ctssdclemn0  7238  exmidfodomrlemeldju  7338  exmidfodomrlemreseldju  7339  pw1on  7372  1tonninf  10623  1dom1el  16126