Theorem 0lt1o 6441

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 2177	. 2 ⊢ ∅ = ∅
2		el1o 6438	. 2 ⊢ (∅ ∈ 1o ↔ ∅ = ∅)
3	1, 2	mpbir 146	1 ⊢ ∅ ∈ 1o

Syntax hints:   = wceq 1353   ∈ wcel 2148  ∅c0 3423  1_oc1o 6410

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-nul 4130

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-dif 3132  df-un 3134  df-nul 3424  df-sn 3599  df-suc 4372  df-1o 6417

This theorem is referenced by:  nnaordex  6529  1domsn  6819  snexxph  6949  difinfsnlem  7098  difinfsn  7099  0ct  7106  ctmlemr  7107  ctssdclemn0  7109  exmidfodomrlemr  7201  exmidfodomrlemrALT  7202  1lt2pi  7339  archnqq  7416  prarloclemarch2  7418  pwle2  14751