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Theorem ondif1 8451
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 8450 . 2 (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ 𝐴 ≠ ∅))
2 on0eln0 6377 . . 3 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
32pm5.32i 576 . 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 397  wcel 2107  wne 2940  cdif 3911  c0 4286  Oncon0 6321  1oc1o 8409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-tr 5227  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-ord 6324  df-on 6325  df-suc 6327  df-1o 8416
This theorem is referenced by:  cantnflem2  9634  oef1o  9642  cnfcom3  9648  infxpenc  9962  onexoegt  41625  ondif1i  41644  omnord1  41687  oenord1  41698  cantnftermord  41702  succlg  41710  dflim5  41711  onmcl  41713  omabs2  41714  naddwordnexlem4  41765
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