Theorem 0lt1o 6584

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

Assertion

Ref	Expression
0lt1o	⊢ ∅ ∈ 1o

Proof of Theorem 0lt1o

Step	Hyp	Ref	Expression
1		eqid 2229	. 2 ⊢ ∅ = ∅
2		el1o 6581	. 2 ⊢ (∅ ∈ 1o ↔ ∅ = ∅)
3	1, 2	mpbir 146	1 ⊢ ∅ ∈ 1o

Syntax hints:   = wceq 1395   ∈ wcel 2200  ∅c0 3491  1_oc1o 6553

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 4209

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 4461  df-1o 6560

This theorem is referenced by:  nnaordex  6672  1domsn  6974  snexxph  7113  difinfsnlem  7262  difinfsn  7263  0ct  7270  ctmlemr  7271  ctssdclemn0  7273  exmidfodomrlemr  7376  exmidfodomrlemrALT  7377  iftrueb01  7404  1lt2pi  7523  archnqq  7600  prarloclemarch2  7602  dom1o  16314  pwle2  16323