Theorem sdomdomtr 7687

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 7580	. . 3 |- ( A ~< B -> A ~<_ B )
2		domtr 7605	. . 3 |- ( ( A ~<_ B /\ B ~<_ C ) -> A ~<_ C )
3	1, 2	sylan 469	. 2 |- ( ( A ~< B /\ B ~<_ C ) -> A ~<_ C )
4		simpl 455	. . 3 |- ( ( A ~< B /\ B ~<_ C ) -> A ~< B )
5		simpr 459	. . . . . 6 |- ( ( A ~< B /\ B ~<_ C ) -> B ~<_ C )
6		ensym 7601	. . . . . 6 |- ( A ~~ C -> C ~~ A )
7		domentr 7611	. . . . . 6 |- ( ( B ~<_ C /\ C ~~ A ) -> B ~<_ A )
8	5, 6, 7	syl2an 475	. . . . 5 |- ( ( ( A ~< B /\ B ~<_ C ) /\ A ~~ C ) -> B ~<_ A )
9		domnsym 7680	. . . . 5 |- ( B ~<_ A -> -. A ~< B )
10	8, 9	syl 17	. . . 4 |- ( ( ( A ~< B /\ B ~<_ C ) /\ A ~~ C ) -> -. A ~< B )
11	10	ex 432	. . 3 |- ( ( A ~< B /\ B ~<_ C ) -> ( A ~~ C -> -. A ~< B ) )
12	4, 11	mt2d 117	. 2 |- ( ( A ~< B /\ B ~<_ C ) -> -. A ~~ C )
13		brsdom 7575	. 2 |- ( A ~< C <-> ( A ~<_ C /\ -. A ~~ C ) )
14	3, 12, 13	sylanbrc 662	1 |- ( ( A ~< B /\ B ~<_ C ) -> A ~< C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   class class class wbr 4394   ~~ cen 7550   ~<_ cdom 7551   ~< csdm 7552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556

This theorem is referenced by:  sdomentr  7688  sucdom  7751  infsdomnn  7814  fodomfib  7833  marypha1lem  7926  r1sdom  8223  infxpenlem  8422  infunsdom1  8624  fin56  8804  fodomb  8935  pwcfsdom  8989  cfpwsdom  8990  canthp1lem2  9060  gchpwdom  9077  gchhar  9086  gchina  9106  tsksdom  9163  tskpr  9177  tskcard  9188  gruina  9225