Theorem sdomdomtr 7702

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 7594	. . 3 |- ( A ~< B -> A ~<_ B )
2		domtr 7619	. . 3 |- ( ( A ~<_ B /\ B ~<_ C ) -> A ~<_ C )
3	1, 2	sylan 474	. 2 |- ( ( A ~< B /\ B ~<_ C ) -> A ~<_ C )
4		simpl 459	. . 3 |- ( ( A ~< B /\ B ~<_ C ) -> A ~< B )
5		simpr 463	. . . . . 6 |- ( ( A ~< B /\ B ~<_ C ) -> B ~<_ C )
6		ensym 7615	. . . . . 6 |- ( A ~~ C -> C ~~ A )
7		domentr 7625	. . . . . 6 |- ( ( B ~<_ C /\ C ~~ A ) -> B ~<_ A )
8	5, 6, 7	syl2an 480	. . . . 5 |- ( ( ( A ~< B /\ B ~<_ C ) /\ A ~~ C ) -> B ~<_ A )
9		domnsym 7695	. . . . 5 |- ( B ~<_ A -> -. A ~< B )
10	8, 9	syl 17	. . . 4 |- ( ( ( A ~< B /\ B ~<_ C ) /\ A ~~ C ) -> -. A ~< B )
11	10	ex 436	. . 3 |- ( ( A ~< B /\ B ~<_ C ) -> ( A ~~ C -> -. A ~< B ) )
12	4, 11	mt2d 121	. 2 |- ( ( A ~< B /\ B ~<_ C ) -> -. A ~~ C )
13		brsdom 7589	. 2 |- ( A ~< C <-> ( A ~<_ C /\ -. A ~~ C ) )
14	3, 12, 13	sylanbrc 669	1 |- ( ( A ~< B /\ B ~<_ C ) -> A ~< C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   class class class wbr 4401   ~~ cen 7563   ~<_ cdom 7564   ~< csdm 7565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569

This theorem is referenced by:  sdomentr  7703  sucdom  7766  infsdomnn  7829  fodomfib  7848  marypha1lem  7944  r1sdom  8242  infxpenlem  8441  infunsdom1  8640  fin56  8820  fodomb  8951  pwcfsdom  9005  cfpwsdom  9006  canthp1lem2  9075  gchpwdom  9092  gchhar  9101  gchina  9121  tsksdom  9178  tskpr  9192  tskcard  9203  gruina  9240