Theorem soirri 6083

Description: A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Hypotheses

Ref	Expression
soi.1	⊢ 𝑅 Or 𝑆
soi.2	⊢ 𝑅 ⊆ (𝑆 × 𝑆)

Assertion

Ref	Expression
soirri	⊢ ¬ 𝐴𝑅𝐴

Proof of Theorem soirri

Step	Hyp	Ref	Expression
1		soi.1	. . . 4 ⊢ 𝑅 Or 𝑆
2		sonr 5556	. . . 4 ⊢ ((𝑅 Or 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴)
3	1, 2	mpan 690	. . 3 ⊢ (𝐴 ∈ 𝑆 → ¬ 𝐴𝑅𝐴)
4	3	adantl 481	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴)
5		soi.2	. . . 4 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
6	5	brel 5689	. . 3 ⊢ (𝐴𝑅𝐴 → (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆))
7	6	con3i 154	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴)
8	4, 7	pm2.61i 182	1 ⊢ ¬ 𝐴𝑅𝐴

Syntax hints:  ¬ wn 3   ∧ wa 395   ∈ wcel 2113   ⊆ wss 3901   class class class wbr 5098   Or wor 5531   × cxp 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-po 5532  df-so 5533  df-xp 5630

This theorem is referenced by:  son2lpi  6085  nqpr  10925  ltapr  10956