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Theorem ondif1 8531
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 8530 . 2 (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ 𝐴 ≠ ∅))
2 on0eln0 6432 . . 3 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
32pm5.32i 573 . 2 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ↔ (𝐴 ∈ On ∧ 𝐴 ≠ ∅))
41, 3bitr4i 277 1 (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  wcel 2099  wne 2930  cdif 3944  c0 4325  Oncon0 6376  1oc1o 8489
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 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-tr 5271  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6379  df-on 6380  df-suc 6382  df-1o 8496
This theorem is referenced by:  cantnflem2  9733  oef1o  9741  cnfcom3  9747  infxpenc  10061  onexoegt  42909  ondif1i  42928  omnord1  42971  oenord1  42982  cantnftermord  42986  succlg  42994  dflim5  42995  onmcl  42997  omabs2  42998  naddwordnexlem4  43068
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