Theorem wesn 5755

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 5754	. . 3 ⊢ (Rel 𝑅 → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
2		sosn 5753	. . 3 ⊢ (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))
3	1, 2	anbi12d 630	. 2 ⊢ (Rel 𝑅 → ((𝑅 Fr {𝐴} ∧ 𝑅 Or {𝐴}) ↔ (¬ 𝐴𝑅𝐴 ∧ ¬ 𝐴𝑅𝐴)))
4		df-we 5624	. 2 ⊢ (𝑅 We {𝐴} ↔ (𝑅 Fr {𝐴} ∧ 𝑅 Or {𝐴}))
5		pm4.24 563	. 2 ⊢ (¬ 𝐴𝑅𝐴 ↔ (¬ 𝐴𝑅𝐴 ∧ ¬ 𝐴𝑅𝐴))
6	3, 4, 5	3bitr4g 314	1 ⊢ (Rel 𝑅 → (𝑅 We {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  {csn 4621   class class class wbr 5139   Or wor 5578   Fr wfr 5619   We wwe 5621  Rel wrel 5672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674

This theorem is referenced by:  0we1  8502  canthwe  10643