MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordge1n0 Structured version   Visualization version   GIF version

Theorem ordge1n0 8492
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 8491 . 2 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))
2 ord0eln0 6412 . 2 (Ord 𝐴 → (∅ ∈ 𝐴𝐴 ≠ ∅))
31, 2bitr3d 281 1 (Ord 𝐴 → (1o𝐴𝐴 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098  wne 2934  wss 3943  c0 4317  Ord word 6356  1oc1o 8457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361  df-suc 6363  df-1o 8464
This theorem is referenced by:  om00  8573  bday1s  27715  finxpsuc  36786  oege1  42613
  Copyright terms: Public domain W3C validator