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Theorem sdomdomtr 8090
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 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Proof of Theorem sdomdomtr
StepHypRef Expression
1 sdomdom 7980 . . 3 (𝐴𝐵𝐴𝐵)
2 domtr 8006 . . 3 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
31, 2sylan 488 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
4 simpl 473 . . 3 ((𝐴𝐵𝐵𝐶) → 𝐴𝐵)
5 simpr 477 . . . . . 6 ((𝐴𝐵𝐵𝐶) → 𝐵𝐶)
6 ensym 8002 . . . . . 6 (𝐴𝐶𝐶𝐴)
7 domentr 8012 . . . . . 6 ((𝐵𝐶𝐶𝐴) → 𝐵𝐴)
85, 6, 7syl2an 494 . . . . 5 (((𝐴𝐵𝐵𝐶) ∧ 𝐴𝐶) → 𝐵𝐴)
9 domnsym 8083 . . . . 5 (𝐵𝐴 → ¬ 𝐴𝐵)
108, 9syl 17 . . . 4 (((𝐴𝐵𝐵𝐶) ∧ 𝐴𝐶) → ¬ 𝐴𝐵)
1110ex 450 . . 3 ((𝐴𝐵𝐵𝐶) → (𝐴𝐶 → ¬ 𝐴𝐵))
124, 11mt2d 131 . 2 ((𝐴𝐵𝐵𝐶) → ¬ 𝐴𝐶)
13 brsdom 7975 . 2 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴𝐶))
143, 12, 13sylanbrc 698 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   class class class wbr 4651  cen 7949  cdom 7950  csdm 7951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955
This theorem is referenced by:  sdomentr  8091  sucdom  8154  infsdomnn  8218  fodomfib  8237  marypha1lem  8336  r1sdom  8634  infxpenlem  8833  infunsdom1  9032  fin56  9212  fodomb  9345  pwcfsdom  9402  cfpwsdom  9403  canthp1lem2  9472  gchpwdom  9489  gchhar  9498  gchina  9518  tsksdom  9575  tskpr  9589  tskcard  9600  gruina  9637  lindsenlbs  33384
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