Theorem domsdomtr 7549

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 7440	. . 3 |- ( B ~< C -> B ~<_ C )
2		domtr 7465	. . 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 7461	. . . . . 6 |- ( A ~~ C -> C ~~ A )
6		simpl 457	. . . . . 6 |- ( ( A ~<_ B /\ B ~< C ) -> A ~<_ B )
7		endomtr 7470	. . . . . 6 |- ( ( C ~~ A /\ A ~<_ B ) -> C ~<_ B )
8	5, 6, 7	syl2anr 478	. . . . 5 |- ( ( ( A ~<_ B /\ B ~< C ) /\ A ~~ C ) -> C ~<_ B )
9		domnsym 7540	. . . . 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 7435	. 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 4393   ~~ cen 7410   ~<_ cdom 7411   ~< csdm 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416

This theorem is referenced by:  ensdomtr  7550  sdomtr  7552  2pwuninel  7569  card2on  7873  tskwe  8224  harval2  8271  prdom2  8277  infxpenlem  8284  alephsucdom  8353  pwsdompw  8477  infunsdom1  8486  fin34  8663  ondomon  8831  cardmin  8832  konigthlem  8836  gchpwdom  8941  gchina  8970  inar1  9046  tskord  9051  tskuni  9054  tskurn  9060  csdfil  19592