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Theorem ondif1 8515
Description: Two ways to say that 𝐴 is a nonzero ordinal number. Lemma 1.10 of [Schloeder] p. 2. (Contributed by Mario Carneiro, 21-May-2015.)
Assertion
Ref Expression
ondif1 (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))

Proof of Theorem ondif1
StepHypRef Expression
1 dif1o 8514 . 2 (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ 𝐴 ≠ ∅))
2 on0eln0 6419 . . 3 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
32pm5.32i 574 . 2 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ↔ (𝐴 ∈ On ∧ 𝐴 ≠ ∅))
41, 3bitr4i 278 1 (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wcel 2099  wne 2935  cdif 3941  c0 4318  Oncon0 6363  1oc1o 8473
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 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
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 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367  df-suc 6369  df-1o 8480
This theorem is referenced by:  cantnflem2  9707  oef1o  9715  cnfcom3  9721  infxpenc  10035  onexoegt  42644  ondif1i  42663  omnord1  42706  oenord1  42717  cantnftermord  42721  succlg  42729  dflim5  42730  onmcl  42732  omabs2  42733  naddwordnexlem4  42803
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