Theorem son2lpi 5040

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 5038	. 2 |- -. A R A
4	1, 2	sotri 5039	. 2 |- ( ( A R B /\ B R A ) -> A R A )
5	3, 4	mto 663	1 |- -. ( A R B /\ B R A )

Syntax hints:   -. wn 3   /\ wa 104   C_ wss 3144   class class class wbr 4018   Or wor 4310   X. cxp 4639

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-po 4311  df-iso 4312  df-xp 4647

This theorem is referenced by:  nqprdisj  7561  ltexprlemdisj  7623  recexprlemdisj  7647  caucvgprlemnkj  7683  caucvgprprlemnkltj  7706  caucvgprprlemnkeqj  7707  caucvgprprlemnjltk  7708