Theorem sosn 5746

Description: Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Assertion

Ref	Expression
sosn	⊢ (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Proof of Theorem sosn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elsni 4601	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
2		elsni 4601	. . . . . . 7 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
3	2	eqcomd 2769	. . . . . 6 ⊢ (𝑦 ∈ {𝐴} → 𝐴 = 𝑦)
4	1, 3	sylan9eq 2818	. . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦)
5	4	3mix2d 1356	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
6	5	rgen2 3206	. . 3 ⊢ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
7		df-so 5567	. . 3 ⊢ (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
8	6, 7	mpbiran2 723	. 2 ⊢ (𝑅 Or {𝐴} ↔ 𝑅 Po {𝐴})
9		posn 5745	. 2 ⊢ (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
10	8, 9	bitrid 286	1 ⊢ (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   ∈ wcel 2147  ∀wral 3080  {csn 4588   class class class wbr 5108   Po wpo 5564   Or wor 5565  Rel wrel 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-po 5566  df-so 5567  df-xp 5664  df-rel 5665

This theorem is used by:  wesn  5748