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Theorem 0lt1o 6139
Description: Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
Assertion
Ref Expression
0lt1o ∅ ∈ 1𝑜

Proof of Theorem 0lt1o
StepHypRef Expression
1 eqid 2085 . 2 ∅ = ∅
2 el1o 6136 . 2 (∅ ∈ 1𝑜 ↔ ∅ = ∅)
31, 2mpbir 144 1 ∅ ∈ 1𝑜
Colors of variables: wff set class
Syntax hints:   = wceq 1287  wcel 1436  c0 3272  1𝑜c1o 6109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-nul 3933
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-dif 2988  df-un 2990  df-nul 3273  df-sn 3431  df-suc 4165  df-1o 6116
This theorem is referenced by:  nnaordex  6219  1domsn  6468  snexxph  6586  exmidfodomrlemr  6749  exmidfodomrlemrALT  6750  1lt2pi  6820  archnqq  6897  prarloclemarch2  6899
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