Theorem soirri 5131

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 4778	. . . 4 |- ( A R A -> ( A e. S /\ A e. S ) )
5	4	simpld 112	. . 3 |- ( A R A -> A e. S )
6		sonr 4414	. . 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 2202   C_ wss 3200   class class class wbr 4088   Or wor 4392   X. cxp 4723

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 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-po 4393  df-iso 4394  df-xp 4731

This theorem is referenced by:  son2lpi  5133  ltsonq  7617  genpdisj  7742  ltposr  7982  axpre-ltirr  8101