Theorem soirri 5077

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 4727	. . . 4 |- ( A R A -> ( A e. S /\ A e. S ) )
5	4	simpld 112	. . 3 |- ( A R A -> A e. S )
6		sonr 4364	. . 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 640	1 |- -. A R A

Syntax hints:   -. wn 3   e. wcel 2176   C_ wss 3166   class class class wbr 4044   Or wor 4342   X. cxp 4673

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-po 4343  df-iso 4344  df-xp 4681

This theorem is referenced by:  son2lpi  5079  ltsonq  7511  genpdisj  7636  ltposr  7876  axpre-ltirr  7995