Theorem 0lt1o 6703

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 2238	. 2 ⊢ ∅ = ∅
2		el1o 6700	. 2 ⊢ (∅ ∈ 1o ↔ ∅ = ∅)
3	1, 2	mpbir 146	1 ⊢ ∅ ∈ 1o

Syntax hints:   = wceq 1402   ∈ wcel 2209  ∅c0 3520  1_oc1o 6670

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-nul 4254

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-sn 3711  df-suc 4511  df-1o 6677

This theorem is referenced by:  nnaordex  6791  modom  7098  1domsn  7105  dom1o  7106  snexxph  7257  difinfsnlem  7429  difinfsn  7430  0ct  7437  ctmlemr  7438  ctssdclemn0  7440  exmidfodomrlemr  7544  exmidfodomrlemrALT  7545  iftrueb01  7572  1lt2pi  7697  archnqq  7774  prarloclemarch2  7776  pwle2  16942