Theorem soirri 5177

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 4822	. . . 4 |- ( A R A -> ( A e. S /\ A e. S ) )
5	4	simpld 112	. . 3 |- ( A R A -> A e. S )
6		sonr 4457	. . 3 |- ( ( R Or S /\ A e. S ) -> -. A R A )
7	2, 5, 6	sylancr 418	. 2 |- ( A R A -> -. A R A )
8	1, 7	pm2.65i 648	1 |- -. A R A

Syntax hints:   -. wn 3   e. wcel 2209   C_ wss 3220   class class class wbr 4125   Or wor 4435   X. cxp 4767

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-po 4436  df-iso 4437  df-xp 4775

This theorem is referenced by:  son2lpi  5179  ltsonq  7755  genpdisj  7880  ltposr  8120  axpre-ltirr  8239