Theorem wesn 5113

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

Step	Hyp	Ref	Expression
1		frsn 5112	. . 3 ⊢ (Rel 𝑅 → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
2		sosn 5111	. . 3 ⊢ (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))
3	1, 2	anbi12d 743	. 2 ⊢ (Rel 𝑅 → ((𝑅 Fr {𝐴} ∧ 𝑅 Or {𝐴}) ↔ (¬ 𝐴𝑅𝐴 ∧ ¬ 𝐴𝑅𝐴)))
4		df-we 4999	. 2 ⊢ (𝑅 We {𝐴} ↔ (𝑅 Fr {𝐴} ∧ 𝑅 Or {𝐴}))
5		pm4.24 673	. 2 ⊢ (¬ 𝐴𝑅𝐴 ↔ (¬ 𝐴𝑅𝐴 ∧ ¬ 𝐴𝑅𝐴))
6	3, 4, 5	3bitr4g 302	1 ⊢ (Rel 𝑅 → (𝑅 We {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  {csn 4125   class class class wbr 4583   Or wor 4958   Fr wfr 4994   We wwe 4996  Rel wrel 5043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045

This theorem is referenced by:  0we1  7473  canthwe  9352