Theorem sdomdomtr 7555

Description: Transitivity of strict dominance and dominance. Theorem 22(iii) of [Suppes] p. 97. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

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

Proof of Theorem sdomdomtr

Step	Hyp	Ref	Expression
1		sdomdom 7448	. . 3 |- ( A ~< B -> A ~<_ B )
2		domtr 7473	. . 3 |- ( ( A ~<_ B /\ B ~<_ C ) -> A ~<_ C )
3	1, 2	sylan 471	. 2 |- ( ( A ~< B /\ B ~<_ C ) -> A ~<_ C )
4		simpl 457	. . 3 |- ( ( A ~< B /\ B ~<_ C ) -> A ~< B )
5		simpr 461	. . . . . 6 |- ( ( A ~< B /\ B ~<_ C ) -> B ~<_ C )
6		ensym 7469	. . . . . 6 |- ( A ~~ C -> C ~~ A )
7		domentr 7479	. . . . . 6 |- ( ( B ~<_ C /\ C ~~ A ) -> B ~<_ A )
8	5, 6, 7	syl2an 477	. . . . 5 |- ( ( ( A ~< B /\ B ~<_ C ) /\ A ~~ C ) -> B ~<_ A )
9		domnsym 7548	. . . . 5 |- ( B ~<_ A -> -. A ~< B )
10	8, 9	syl 16	. . . 4 |- ( ( ( A ~< B /\ B ~<_ C ) /\ A ~~ C ) -> -. A ~< B )
11	10	ex 434	. . 3 |- ( ( A ~< B /\ B ~<_ C ) -> ( A ~~ C -> -. A ~< B ) )
12	4, 11	mt2d 117	. 2 |- ( ( A ~< B /\ B ~<_ C ) -> -. A ~~ C )
13		brsdom 7443	. 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 4401   ~~ cen 7418   ~<_ cdom 7419   ~< csdm 7420

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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483

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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424

This theorem is referenced by:  sdomentr  7556  sucdom  7620  sucdomiOLD  7621  infsdomnn  7685  fodomfib  7703  marypha1lem  7795  r1sdom  8093  infxpenlem  8292  infunsdom1  8494  fin56  8674  fodomb  8805  pwcfsdom  8859  cfpwsdom  8860  canthp1lem2  8932  gchpwdom  8949  gchhar  8958  gchina  8978  tsksdom  9035  tskpr  9049  tskcard  9060  gruina  9097