Theorem po2nr 4294

Description: A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)

Assertion

Ref	Expression
po2nr	|- ( ( R Po A /\ ( B e. A /\ C e. A ) ) -> -. ( B R C /\ C R B ) )

Proof of Theorem po2nr

Step	Hyp	Ref	Expression
1		poirr 4292	. . 3 |- ( ( R Po A /\ B e. A ) -> -. B R B )
2	1	adantrr 476	. 2 |- ( ( R Po A /\ ( B e. A /\ C e. A ) ) -> -. B R B )
3		potr 4293	. . . . . 6 |- ( ( R Po A /\ ( B e. A /\ C e. A /\ B e. A ) ) -> ( ( B R C /\ C R B ) -> B R B ) )
4	3	3exp2 1220	. . . . 5 |- ( R Po A -> ( B e. A -> ( C e. A -> ( B e. A -> ( ( B R C /\ C R B ) -> B R B ) ) ) ) )
5	4	com34 83	. . . 4 |- ( R Po A -> ( B e. A -> ( B e. A -> ( C e. A -> ( ( B R C /\ C R B ) -> B R B ) ) ) ) )
6	5	pm2.43d 50	. . 3 |- ( R Po A -> ( B e. A -> ( C e. A -> ( ( B R C /\ C R B ) -> B R B ) ) ) )
7	6	imp32 255	. 2 |- ( ( R Po A /\ ( B e. A /\ C e. A ) ) -> ( ( B R C /\ C R B ) -> B R B ) )
8	2, 7	mtod 658	1 |- ( ( R Po A /\ ( B e. A /\ C e. A ) ) -> -. ( B R C /\ C R B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 2141   class class class wbr 3989   Po wpo 4279

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-po 4281

This theorem is referenced by:  po3nr  4295  so2nr  4306  tridc  6877