Theorem son2lpi 5088

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 5086	. 2 |- -. A R A
4	1, 2	sotri 5087	. 2 |- ( ( A R B /\ B R A ) -> A R A )
5	3, 4	mto 664	1 |- -. ( A R B /\ B R A )

Syntax hints:   -. wn 3   /\ wa 104   C_ wss 3170   class class class wbr 4051   Or wor 4350   X. cxp 4681

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-po 4351  df-iso 4352  df-xp 4689

This theorem is referenced by:  nqprdisj  7677  ltexprlemdisj  7739  recexprlemdisj  7763  caucvgprlemnkj  7799  caucvgprprlemnkltj  7822  caucvgprprlemnkeqj  7823  caucvgprprlemnjltk  7824