Theorem son2lpi 5184

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 5182	. 2 |- -. A R A
4	1, 2	sotri 5183	. 2 |- ( ( A R B /\ B R A ) -> A R A )
5	3, 4	mto 672	1 |- -. ( A R B /\ B R A )

This proof depends on syntax axioms:   -. wn 3   /\ wa 104   C_ wss 3220   class class class wbr 4130   Or wor 4440   X. cxp 4772

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-po 4441  df-iso 4442  df-xp 4780

This theorem is used by:  nqprdisj  7911  ltexprlemdisj  7973  recexprlemdisj  7997  caucvgprlemnkj  8033  caucvgprprlemnkltj  8056  caucvgprprlemnkeqj  8057  caucvgprprlemnjltk  8058