Theorem 0lt1o 8471

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 2730	. 2 ⊢ ∅ = ∅
2		el1o 8462	. 2 ⊢ (∅ ∈ 1o ↔ ∅ = ∅)
3	1, 2	mpbir 231	1 ⊢ ∅ ∈ 1o

Syntax hints:   = wceq 1540   ∈ wcel 2109  ∅c0 4299  1_oc1o 8430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3920  df-un 3922  df-nul 4300  df-sn 4593  df-suc 6341  df-1o 8437

This theorem is referenced by:  dif20el  8472  oe1m  8512  oen0  8553  oeoa  8564  oeoe  8566  isfin4p1  10275  fin1a2lem4  10363  1lt2pi  10865  indpi  10867  sadcp1  16432  vr1cl2  22084  fvcoe1  22099  vr1cl  22109  subrgvr1cl  22155  coe1mul2lem1  22160  coe1tm  22166  ply1coe  22192  evl1var  22230  evls1var  22232  rhmply1vr1  22281  xkofvcn  23578  pw2f1ocnv  43033  wepwsolem  43038  onexoegt  43240  oaordnrex  43291  omnord1ex  43300  omcl3g  43330  tfsconcatb0  43340  indthinc  49455  indthincALT  49456  prsthinc  49457  setc1oid  49488  funcsetc1ocl  49489  funcsetc1o  49490  isinito2lem  49491  isinito4  49540  setc1onsubc  49595