Theorem 0lt1o 6603

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 2229	. 2 ⊢ ∅ = ∅
2		el1o 6600	. 2 ⊢ (∅ ∈ 1o ↔ ∅ = ∅)
3	1, 2	mpbir 146	1 ⊢ ∅ ∈ 1o

Syntax hints:   = wceq 1395   ∈ wcel 2200  ∅c0 3492  1_oc1o 6570

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-nul 4213

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-dif 3200  df-un 3202  df-nul 3493  df-sn 3673  df-suc 4466  df-1o 6577

This theorem is referenced by:  nnaordex  6691  modom  6989  1domsn  6996  dom1o  6997  snexxph  7140  difinfsnlem  7289  difinfsn  7290  0ct  7297  ctmlemr  7298  ctssdclemn0  7300  exmidfodomrlemr  7403  exmidfodomrlemrALT  7404  iftrueb01  7431  1lt2pi  7550  archnqq  7627  prarloclemarch2  7629  pwle2  16535