Theorem el1o 6523

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 6515	. . 3 |- 1o = { (/) }
2	1	eleq2i 2272	. 2 |- ( A e. 1o <-> A e. { (/) } )
3		0ex 4171	. . 3 |- (/) e. _V
4	3	elsn2 3667	. 2 |- ( A e. { (/) } <-> A = (/) )
5	2, 4	bitri 184	1 |- ( A e. 1o <-> A = (/) )

Syntax hints:   <-> wb 105   = wceq 1373   e. wcel 2176   (/)c0 3460   {csn 3633   1oc1o 6495

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-nul 4170

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-un 3170  df-nul 3461  df-sn 3639  df-suc 4418  df-1o 6502

This theorem is referenced by:  0lt1o  6526  map0e  6773  map1  6904  omp1eomlem  7196  ctmlemr  7210  ctssdclemn0  7212  exmidfodomrlemeldju  7307  exmidfodomrlemreseldju  7308  pw1on  7338  1tonninf  10586  1dom1el  15927