Theorem 0lt1o 6495

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 2193	. 2 ⊢ ∅ = ∅
2		el1o 6492	. 2 ⊢ (∅ ∈ 1o ↔ ∅ = ∅)
3	1, 2	mpbir 146	1 ⊢ ∅ ∈ 1o

Syntax hints:   = wceq 1364   ∈ wcel 2164  ∅c0 3447  1_oc1o 6464

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-nul 4156

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3156  df-un 3158  df-nul 3448  df-sn 3625  df-suc 4403  df-1o 6471

This theorem is referenced by:  nnaordex  6583  1domsn  6875  snexxph  7011  difinfsnlem  7160  difinfsn  7161  0ct  7168  ctmlemr  7169  ctssdclemn0  7171  exmidfodomrlemr  7264  exmidfodomrlemrALT  7265  1lt2pi  7402  archnqq  7479  prarloclemarch2  7481  pwle2  15559