Theorem soirri 6073

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 5548	. . . 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 5681	. . 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 2111   ⊆ wss 3902   class class class wbr 5091   Or wor 5523   × cxp 5614

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

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 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-po 5524  df-so 5525  df-xp 5622

This theorem is referenced by:  son2lpi  6075  nqpr  10902  ltapr  10933