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Theorem ondif1 8319
Description: Two ways to say that 𝐴 is a nonzero ordinal number. (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 8318 . 2 (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ 𝐴 ≠ ∅))
2 on0eln0 6315 . . 3 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
32pm5.32i 575 . 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 396  wcel 2106  wne 2943  cdif 3884  c0 4257  Oncon0 6260  1oc1o 8278
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5491  df-po 5499  df-so 5500  df-fr 5540  df-we 5542  df-ord 6263  df-on 6264  df-suc 6266  df-1o 8285
This theorem is referenced by:  cantnflem2  9436  oef1o  9444  cnfcom3  9450  infxpenc  9762
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