Theorem soirri 5522

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 5056	. . . 4 ⊢ ((𝑅 Or 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴)
3	1, 2	mpan 706	. . 3 ⊢ (𝐴 ∈ 𝑆 → ¬ 𝐴𝑅𝐴)
4	3	adantl 482	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴)
5		soi.2	. . . 4 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
6	5	brel 5168	. . 3 ⊢ (𝐴𝑅𝐴 → (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆))
7	6	con3i 150	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴)
8	4, 7	pm2.61i 176	1 ⊢ ¬ 𝐴𝑅𝐴

Syntax hints:  ¬ wn 3   ∧ wa 384   ∈ wcel 1990   ⊆ wss 3574   class class class wbr 4653   Or wor 5034   × cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-xp 5120

This theorem is referenced by:  son2lpi  5524  nqpr  9836  ltapr  9867