Theorem soirri 6079

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 5555	. . . 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 5688	. . 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 2109   ⊆ wss 3905   class class class wbr 5095   Or wor 5530   × cxp 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-po 5531  df-so 5532  df-xp 5629

This theorem is referenced by:  son2lpi  6081  nqpr  10927  ltapr  10958