Theorem 0lt1o 7852

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 2826	. 2 ⊢ ∅ = ∅
2		el1o 7847	. 2 ⊢ (∅ ∈ 1o ↔ ∅ = ∅)
3	1, 2	mpbir 223	1 ⊢ ∅ ∈ 1o

Syntax hints:   = wceq 1658   ∈ wcel 2166  ∅c0 4145  1_oc1o 7820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-nul 5014

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-v 3417  df-dif 3802  df-un 3804  df-nul 4146  df-sn 4399  df-suc 5970  df-1o 7827

This theorem is referenced by:  dif20el  7853  oe1m  7893  oen0  7934  oeoa  7945  oeoe  7947  isfin4-3  9453  fin1a2lem4  9541  1lt2pi  10043  indpi  10045  sadcp1  15551  vr1cl2  19924  fvcoe1  19938  vr1cl  19948  subrgvr1cl  19993  coe1mul2lem1  19998  coe1tm  20004  ply1coe  20027  evl1var  20061  evls1var  20063  xkofvcn  21859  pw2f1ocnv  38448  wepwsolem  38456