Theorem 0lt1o 6651

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 2231	. 2 ⊢ ∅ = ∅
2		el1o 6648	. 2 ⊢ (∅ ∈ 1o ↔ ∅ = ∅)
3	1, 2	mpbir 146	1 ⊢ ∅ ∈ 1o

Syntax hints:   = wceq 1398   ∈ wcel 2202  ∅c0 3496  1_oc1o 6618

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 2213  ax-nul 4220

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-un 3205  df-nul 3497  df-sn 3679  df-suc 4474  df-1o 6625

This theorem is referenced by:  nnaordex  6739  modom  7037  1domsn  7044  dom1o  7045  snexxph  7192  difinfsnlem  7341  difinfsn  7342  0ct  7349  ctmlemr  7350  ctssdclemn0  7352  exmidfodomrlemr  7456  exmidfodomrlemrALT  7457  iftrueb01  7484  1lt2pi  7603  archnqq  7680  prarloclemarch2  7682  pwle2  16703