Theorem soirri 6136

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 5588	. . . 4 ⊢ ((𝑅 Or 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴)
3	1, 2	mpan 703	. . 3 ⊢ (𝐴 ∈ 𝑆 → ¬ 𝐴𝑅𝐴)
4	3	adantl 487	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴)
5		soi.2	. . . 4 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
6	5	brel 5721	. . 3 ⊢ (𝐴𝑅𝐴 → (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆))
7	6	con3i 155	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴)
8	4, 7	pm2.61i 184	1 ⊢ ¬ 𝐴𝑅𝐴

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∈ wcel 2147   ⊆ wss 3900   class class class wbr 5106   Or wor 5563   × cxp 5654

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 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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 4839  df-br 5107  df-opab 5172  df-po 5564  df-so 5565  df-xp 5662

This theorem is used by:  son2lpi  6138  nqpr  11136  ltapr  11167