Theorem el1o 6583

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 6575	. . 3 |- 1o = { (/) }
2	1	eleq2i 2296	. 2 |- ( A e. 1o <-> A e. { (/) } )
3		0ex 4211	. . 3 |- (/) e. _V
4	3	elsn2 3700	. 2 |- ( A e. { (/) } <-> A = (/) )
5	2, 4	bitri 184	1 |- ( A e. 1o <-> A = (/) )

Syntax hints:   <-> wb 105   = wceq 1395   e. wcel 2200   (/)c0 3491   {csn 3666   1oc1o 6555

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-nul 4210

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-nul 3492  df-sn 3672  df-suc 4462  df-1o 6562

This theorem is referenced by:  0lt1o  6586  map0e  6833  map1  6965  omp1eomlem  7261  ctmlemr  7275  ctssdclemn0  7277  exmidfodomrlemeldju  7377  exmidfodomrlemreseldju  7378  pw1on  7411  1tonninf  10663  1dom1el  16354