Theorem soirri 5133

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		id 19	. 2 ⊢ (𝐴𝑅𝐴 → 𝐴𝑅𝐴)
2		soi.1	. . 3 ⊢ 𝑅 Or 𝑆
3		soi.2	. . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
4	3	brel 4780	. . . 4 ⊢ (𝐴𝑅𝐴 → (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆))
5	4	simpld 112	. . 3 ⊢ (𝐴𝑅𝐴 → 𝐴 ∈ 𝑆)
6		sonr 4416	. . 3 ⊢ ((𝑅 Or 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴)
7	2, 5, 6	sylancr 414	. 2 ⊢ (𝐴𝑅𝐴 → ¬ 𝐴𝑅𝐴)
8	1, 7	pm2.65i 644	1 ⊢ ¬ 𝐴𝑅𝐴

Syntax hints:  ¬ wn 3   ∈ wcel 2201   ⊆ wss 3199   class class class wbr 4089   Or wor 4394   × cxp 4725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-br 4090  df-opab 4152  df-po 4395  df-iso 4396  df-xp 4733

This theorem is referenced by:  son2lpi  5135  ltsonq  7623  genpdisj  7748  ltposr  7988  axpre-ltirr  8107