Theorem 0lt1o 6539

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 2206	. 2 ⊢ ∅ = ∅
2		el1o 6536	. 2 ⊢ (∅ ∈ 1o ↔ ∅ = ∅)
3	1, 2	mpbir 146	1 ⊢ ∅ ∈ 1o

Syntax hints:   = wceq 1373   ∈ wcel 2177  ∅c0 3464  1_oc1o 6508

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-nul 4178

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3172  df-un 3174  df-nul 3465  df-sn 3644  df-suc 4426  df-1o 6515

This theorem is referenced by:  nnaordex  6627  1domsn  6929  snexxph  7067  difinfsnlem  7216  difinfsn  7217  0ct  7224  ctmlemr  7225  ctssdclemn0  7227  exmidfodomrlemr  7326  exmidfodomrlemrALT  7327  iftrueb01  7354  1lt2pi  7473  archnqq  7550  prarloclemarch2  7552  dom1o  16067  pwle2  16076