Theorem el1o 6670

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 6661	. . 3 |- 1o = { (/) }
2	1	eleq2i 2299	. 2 |- ( A e. 1o <-> A e. { (/) } )
3		0ex 4237	. . 3 |- (/) e. _V
4	3	elsn2 3723	. 2 |- ( A e. { (/) } <-> A = (/) )
5	2, 4	bitri 184	1 |- ( A e. 1o <-> A = (/) )

Syntax hints:   <-> wb 105   = wceq 1398   e. wcel 2203   (/)c0 3508   {csn 3689   1oc1o 6640

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-nul 4236

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-dif 3213  df-un 3215  df-nul 3509  df-sn 3695  df-suc 4492  df-1o 6647

This theorem is referenced by:  0lt1o  6673  map0e  6920  map1  7054  1dom1el  7060  omp1eomlem  7385  ctmlemr  7399  ctssdclemn0  7401  exmidfodomrlemeldju  7502  exmidfodomrlemreseldju  7503  pw1on  7536  1tonninf  10803