Theorem 0lt1o 7471

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 2610	. 2 ⊢ ∅ = ∅
2		el1o 7466	. 2 ⊢ (∅ ∈ 1𝑜 ↔ ∅ = ∅)
3	1, 2	mpbir 220	1 ⊢ ∅ ∈ 1𝑜

Syntax hints:   = wceq 1475   ∈ wcel 1977  ∅c0 3874  1_𝑜c1o 7440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-suc 5646  df-1o 7447

This theorem is referenced by:  dif20el  7472  oe1m  7512  oen0  7553  oeoa  7564  oeoe  7566  isfin4-3  9020  fin1a2lem4  9108  1lt2pi  9606  indpi  9608  sadcp1  15015  vr1cl2  19384  fvcoe1  19398  vr1cl  19408  subrgvr1cl  19453  coe1mul2lem1  19458  coe1tm  19464  ply1coe  19487  evl1var  19521  evls1var  19523  xkofvcn  21297  pw2f1ocnv  36622  wepwsolem  36630