Theorem 0lt1o 6586

Description: Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)

Assertion

Ref	Expression
0lt1o	|- (/) e. 1o

Proof of Theorem 0lt1o

Step	Hyp	Ref	Expression
1		eqid 2229	. 2 |- (/) = (/)
2		el1o 6583	. 2 |- ( (/) e. 1o <-> (/) = (/) )
3	1, 2	mpbir 146	1 |- (/) e. 1o

Syntax hints:   = wceq 1395   e. wcel 2200   (/)c0 3491   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:  nnaordex  6674  1domsn  6976  dom1o  6977  snexxph  7117  difinfsnlem  7266  difinfsn  7267  0ct  7274  ctmlemr  7275  ctssdclemn0  7277  exmidfodomrlemr  7380  exmidfodomrlemrALT  7381  iftrueb01  7408  1lt2pi  7527  archnqq  7604  prarloclemarch2  7606  pwle2  16364