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Theorem wesn 5764
Description: Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
Assertion
Ref Expression
wesn (Rel 𝑅 → (𝑅 We {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Proof of Theorem wesn
StepHypRef Expression
1 frsn 5763 . . 3 (Rel 𝑅 → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
2 sosn 5762 . . 3 (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))
31, 2anbi12d 631 . 2 (Rel 𝑅 → ((𝑅 Fr {𝐴} ∧ 𝑅 Or {𝐴}) ↔ (¬ 𝐴𝑅𝐴 ∧ ¬ 𝐴𝑅𝐴)))
4 df-we 5633 . 2 (𝑅 We {𝐴} ↔ (𝑅 Fr {𝐴} ∧ 𝑅 Or {𝐴}))
5 pm4.24 564 . 2 𝐴𝑅𝐴 ↔ (¬ 𝐴𝑅𝐴 ∧ ¬ 𝐴𝑅𝐴))
63, 4, 53bitr4g 313 1 (Rel 𝑅 → (𝑅 We {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  {csn 4628   class class class wbr 5148   Or wor 5587   Fr wfr 5628   We wwe 5630  Rel wrel 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683
This theorem is referenced by:  0we1  8505  canthwe  10645
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