Theorem el1o 6434

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 6426	. . 3 |- 1o = { (/) }
2	1	eleq2i 2244	. 2 |- ( A e. 1o <-> A e. { (/) } )
3		0ex 4129	. . 3 |- (/) e. _V
4	3	elsn2 3626	. 2 |- ( A e. { (/) } <-> A = (/) )
5	2, 4	bitri 184	1 |- ( A e. 1o <-> A = (/) )

Syntax hints:   <-> wb 105   = wceq 1353   e. wcel 2148   (/)c0 3422   {csn 3592   1oc1o 6406

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-nul 4128

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-un 3133  df-nul 3423  df-sn 3598  df-suc 4370  df-1o 6413

This theorem is referenced by:  0lt1o  6437  map0e  6682  map1  6808  omp1eomlem  7089  ctmlemr  7103  ctssdclemn0  7105  exmidfodomrlemeldju  7194  exmidfodomrlemreseldju  7195  pw1on  7221  1tonninf  10434