Theorem 0lt1o 6419

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 2170	. 2 |- (/) = (/)
2		el1o 6416	. 2 |- ( (/) e. 1o <-> (/) = (/) )
3	1, 2	mpbir 145	1 |- (/) e. 1o

Syntax hints:   = wceq 1348   e. wcel 2141   (/)c0 3414   1oc1o 6388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4115

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-nul 3415  df-sn 3589  df-suc 4356  df-1o 6395

This theorem is referenced by:  nnaordex  6507  1domsn  6797  snexxph  6927  difinfsnlem  7076  difinfsn  7077  0ct  7084  ctmlemr  7085  ctssdclemn0  7087  exmidfodomrlemr  7179  exmidfodomrlemrALT  7180  1lt2pi  7302  archnqq  7379  prarloclemarch2  7381  pwle2  14031