Theorem el1o 6342

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 6334	. . 3 |- 1o = { (/) }
2	1	eleq2i 2207	. 2 |- ( A e. 1o <-> A e. { (/) } )
3		0ex 4063	. . 3 |- (/) e. _V
4	3	elsn2 3566	. 2 |- ( A e. { (/) } <-> A = (/) )
5	2, 4	bitri 183	1 |- ( A e. 1o <-> A = (/) )

Syntax hints:   <-> wb 104   = wceq 1332   e. wcel 1481   (/)c0 3368   {csn 3532   1oc1o 6314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4062

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-un 3080  df-nul 3369  df-sn 3538  df-suc 4301  df-1o 6321

This theorem is referenced by:  0lt1o  6345  map0e  6588  map1  6714  omp1eomlem  6987  ctmlemr  7001  ctssdclemn0  7003  exmidfodomrlemeldju  7072  exmidfodomrlemreseldju  7073  1tonninf  10244