Theorem 0lt1o 6673

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 2232	. 2 ⊢ ∅ = ∅
2		el1o 6670	. 2 ⊢ (∅ ∈ 1o ↔ ∅ = ∅)
3	1, 2	mpbir 146	1 ⊢ ∅ ∈ 1o

Syntax hints:   = wceq 1398   ∈ wcel 2203  ∅c0 3508  1_oc1o 6640

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 2214  ax-nul 4236

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-dif 3213  df-un 3215  df-nul 3509  df-sn 3695  df-suc 4492  df-1o 6647

This theorem is referenced by:  nnaordex  6761  modom  7061  1domsn  7068  dom1o  7069  snexxph  7220  difinfsnlem  7390  difinfsn  7391  0ct  7398  ctmlemr  7399  ctssdclemn0  7401  exmidfodomrlemr  7505  exmidfodomrlemrALT  7506  iftrueb01  7533  1lt2pi  7655  archnqq  7732  prarloclemarch2  7734  pwle2  16772