Theorem soirri 5162

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 C_ ( S X. S )

Assertion

Ref	Expression
soirri	|- -. A R A

Proof of Theorem soirri

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( A R A -> A R A )
2		soi.1	. . 3 |- R Or S
3		soi.2	. . . . 5 |- R C_ ( S X. S )
4	3	brel 4807	. . . 4 |- ( A R A -> ( A e. S /\ A e. S ) )
5	4	simpld 112	. . 3 |- ( A R A -> A e. S )
6		sonr 4443	. . 3 |- ( ( R Or S /\ A e. S ) -> -. A R A )
7	2, 5, 6	sylancr 414	. 2 |- ( A R A -> -. A R A )
8	1, 7	pm2.65i 644	1 |- -. A R A

Syntax hints:   -. wn 3   e. wcel 2205   C_ wss 3214   class class class wbr 4114   Or wor 4421   X. cxp 4752

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-po 4422  df-iso 4423  df-xp 4760

This theorem is referenced by:  son2lpi  5164  ltsonq  7729  genpdisj  7854  ltposr  8094  axpre-ltirr  8213