Theorem el1o 6490

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 6482	. . 3 |- 1o = { (/) }
2	1	eleq2i 2260	. 2 |- ( A e. 1o <-> A e. { (/) } )
3		0ex 4156	. . 3 |- (/) e. _V
4	3	elsn2 3652	. 2 |- ( A e. { (/) } <-> A = (/) )
5	2, 4	bitri 184	1 |- ( A e. 1o <-> A = (/) )

Syntax hints:   <-> wb 105   = wceq 1364   e. wcel 2164   (/)c0 3446   {csn 3618   1oc1o 6462

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-nul 4155

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-un 3157  df-nul 3447  df-sn 3624  df-suc 4402  df-1o 6469

This theorem is referenced by:  0lt1o  6493  map0e  6740  map1  6866  omp1eomlem  7153  ctmlemr  7167  ctssdclemn0  7169  exmidfodomrlemeldju  7259  exmidfodomrlemreseldju  7260  pw1on  7286  1tonninf  10512  1dom1el  15483