Theorem 0lt1o 6608

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 2231	. 2 ⊢ ∅ = ∅
2		el1o 6605	. 2 ⊢ (∅ ∈ 1o ↔ ∅ = ∅)
3	1, 2	mpbir 146	1 ⊢ ∅ ∈ 1o

Syntax hints:   = wceq 1397   ∈ wcel 2202  ∅c0 3494  1_oc1o 6575

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-nul 4215

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-un 3204  df-nul 3495  df-sn 3675  df-suc 4468  df-1o 6582

This theorem is referenced by:  nnaordex  6696  modom  6994  1domsn  7001  dom1o  7002  snexxph  7149  difinfsnlem  7298  difinfsn  7299  0ct  7306  ctmlemr  7307  ctssdclemn0  7309  exmidfodomrlemr  7413  exmidfodomrlemrALT  7414  iftrueb01  7441  1lt2pi  7560  archnqq  7637  prarloclemarch2  7639  pwle2  16625