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Theorem ordge1n0 8531
Description: An ordinal greater than or equal to 1 is nonzero. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
ordge1n0 (Ord 𝐴 → (1o𝐴𝐴 ≠ ∅))

Proof of Theorem ordge1n0
StepHypRef Expression
1 ordgt0ge1 8530 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))
2 ord0eln0 6441 . 2 (Ord 𝐴 → (∅ ∈ 𝐴𝐴 ≠ ∅))
31, 2bitr3d 281 1 (Ord 𝐴 → (1o𝐴𝐴 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2106  wne 2938  wss 3963  c0 4339  Ord word 6385  1oc1o 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392  df-1o 8505
This theorem is referenced by:  om00  8612  bday1s  27891  finxpsuc  37381  oege1  43296
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