Theorem son2lpi 5078

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 5076	. 2 |- -. A R A
4	1, 2	sotri 5077	. 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 3165   class class class wbr 4043   Or wor 4341   X. cxp 4672

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-po 4342  df-iso 4343  df-xp 4680

This theorem is referenced by:  nqprdisj  7656  ltexprlemdisj  7718  recexprlemdisj  7742  caucvgprlemnkj  7778  caucvgprprlemnkltj  7801  caucvgprprlemnkeqj  7802  caucvgprprlemnjltk  7803