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Theorem ordge1n0 8515
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 8514 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))
2 ord0eln0 6424 . 2 (Ord 𝐴 → (∅ ∈ 𝐴𝐴 ≠ ∅))
31, 2bitr3d 281 1 (Ord 𝐴 → (1o𝐴𝐴 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2099  wne 2937  wss 3947  c0 4323  Ord word 6368  1oc1o 8480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6372  df-on 6373  df-suc 6375  df-1o 8487
This theorem is referenced by:  om00  8596  bday1s  27777  finxpsuc  36877  oege1  42735
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