Theorem son2lpi 4930

Description: A strict order relation has no 2-cycle loops. (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
son2lpi	|- -. ( A R B /\ B R A )

Proof of Theorem son2lpi

Step	Hyp	Ref	Expression
1		soi.1	. . 3 |- R Or S
2		soi.2	. . 3 |- R C_ ( S X. S )
3	1, 2	soirri 4928	. 2 |- -. A R A
4	1, 2	sotri 4929	. 2 |- ( ( A R B /\ B R A ) -> A R A )
5	3, 4	mto 651	1 |- -. ( A R B /\ B R A )

Syntax hints:   -. wn 3   /\ wa 103   C_ wss 3066   class class class wbr 3924   Or wor 4212   X. cxp 4532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-po 4213  df-iso 4214  df-xp 4540

This theorem is referenced by:  nqprdisj  7345  ltexprlemdisj  7407  recexprlemdisj  7431  caucvgprlemnkj  7467  caucvgprprlemnkltj  7490  caucvgprprlemnkeqj  7491  caucvgprprlemnjltk  7492