Theorem sdomdomtr 7640

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 7533	. . 3 |- ( A ~< B -> A ~<_ B )
2		domtr 7558	. . 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 7554	. . . . . 6 |- ( A ~~ C -> C ~~ A )
7		domentr 7564	. . . . . 6 |- ( ( B ~<_ C /\ C ~~ A ) -> B ~<_ A )
8	5, 6, 7	syl2an 477	. . . . 5 |- ( ( ( A ~< B /\ B ~<_ C ) /\ A ~~ C ) -> B ~<_ A )
9		domnsym 7633	. . . . 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 7528	. 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 4440   ~~ cen 7503   ~<_ cdom 7504   ~< csdm 7505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509

This theorem is referenced by:  sdomentr  7641  sucdom  7705  sucdomiOLD  7706  infsdomnn  7770  fodomfib  7789  marypha1lem  7882  r1sdom  8181  infxpenlem  8380  infunsdom1  8582  fin56  8762  fodomb  8893  pwcfsdom  8947  cfpwsdom  8948  canthp1lem2  9020  gchpwdom  9037  gchhar  9046  gchina  9066  tsksdom  9123  tskpr  9137  tskcard  9148  gruina  9185