Theorem 0lt1o 6267

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 2100	. 2 ⊢ ∅ = ∅
2		el1o 6264	. 2 ⊢ (∅ ∈ 1o ↔ ∅ = ∅)
3	1, 2	mpbir 145	1 ⊢ ∅ ∈ 1o

Syntax hints:   = wceq 1299   ∈ wcel 1448  ∅c0 3310  1_oc1o 6236

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-nul 3994

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-dif 3023  df-un 3025  df-nul 3311  df-sn 3480  df-suc 4231  df-1o 6243

This theorem is referenced by:  nnaordex  6353  1domsn  6642  snexxph  6766  difinfsnlem  6899  difinfsn  6900  0ct  6907  ctmlemr  6908  ctssdclemn0  6910  exmidfodomrlemr  6967  exmidfodomrlemrALT  6968  1lt2pi  7049  archnqq  7126  prarloclemarch2  7128  pwle2  12779