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Theorem 0lt1o 6023
 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 2040 . 2 ∅ = ∅
2 el1o 6020 . 2 (∅ ∈ 1𝑜 ↔ ∅ = ∅)
31, 2mpbir 134 1 ∅ ∈ 1𝑜
 Colors of variables: wff set class Syntax hints:   = wceq 1243   ∈ wcel 1393  ∅c0 3224  1𝑜c1o 5994 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-nul 3225  df-sn 3381  df-suc 4108  df-1o 6001 This theorem is referenced by:  nnaordex  6100  1lt2pi  6438  archnqq  6515  prarloclemarch2  6517
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