Theorem domsdomtr 7654

Description: Transitivity of dominance and strict dominance. Theorem 22(ii) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
domsdomtr	|- ( ( A ~<_ B /\ B ~< C ) -> A ~< C )

Proof of Theorem domsdomtr

Step	Hyp	Ref	Expression
1		sdomdom 7545	. . 3 |- ( B ~< C -> B ~<_ C )
2		domtr 7570	. . 3 |- ( ( A ~<_ B /\ B ~<_ C ) -> A ~<_ C )
3	1, 2	sylan2 474	. 2 |- ( ( A ~<_ B /\ B ~< C ) -> A ~<_ C )
4		simpr 461	. . 3 |- ( ( A ~<_ B /\ B ~< C ) -> B ~< C )
5		ensym 7566	. . . . . 6 |- ( A ~~ C -> C ~~ A )
6		simpl 457	. . . . . 6 |- ( ( A ~<_ B /\ B ~< C ) -> A ~<_ B )
7		endomtr 7575	. . . . . 6 |- ( ( C ~~ A /\ A ~<_ B ) -> C ~<_ B )
8	5, 6, 7	syl2anr 478	. . . . 5 |- ( ( ( A ~<_ B /\ B ~< C ) /\ A ~~ C ) -> C ~<_ B )
9		domnsym 7645	. . . . 5 |- ( C ~<_ B -> -. B ~< C )
10	8, 9	syl 16	. . . 4 |- ( ( ( A ~<_ B /\ B ~< C ) /\ A ~~ C ) -> -. B ~< C )
11	10	ex 434	. . 3 |- ( ( A ~<_ B /\ B ~< C ) -> ( A ~~ C -> -. B ~< C ) )
12	4, 11	mt2d 117	. 2 |- ( ( A ~<_ B /\ B ~< C ) -> -. A ~~ C )
13		brsdom 7540	. 2 |- ( A ~< C <-> ( A ~<_ C /\ -. A ~~ C ) )
14	3, 12, 13	sylanbrc 664	1 |- ( ( A ~<_ B /\ B ~< C ) -> A ~< C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   class class class wbr 4437   ~~ cen 7515   ~<_ cdom 7516   ~< csdm 7517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521

This theorem is referenced by:  ensdomtr  7655  sdomtr  7657  2pwuninel  7674  card2on  7983  tskwe  8334  harval2  8381  prdom2  8387  infxpenlem  8394  alephsucdom  8463  pwsdompw  8587  infunsdom1  8596  fin34  8773  ondomon  8941  cardmin  8942  konigthlem  8946  gchpwdom  9051  gchina  9080  inar1  9156  tskord  9161  tskuni  9164  tskurn  9170  csdfil  20373