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Theorem sdomdomtr 7226
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
StepHypRef Expression
1 sdomdom 7121 . . 3  |-  ( A 
~<  B  ->  A  ~<_  B )
2 domtr 7146 . . 3  |-  ( ( A  ~<_  B  /\  B  ~<_  C )  ->  A  ~<_  C )
31, 2sylan 458 . 2  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  A  ~<_  C )
4 simpl 444 . . 3  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  A  ~<  B )
5 simpr 448 . . . . . 6  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  B  ~<_  C )
6 ensym 7142 . . . . . 6  |-  ( A 
~~  C  ->  C  ~~  A )
7 domentr 7152 . . . . . 6  |-  ( ( B  ~<_  C  /\  C  ~~  A )  ->  B  ~<_  A )
85, 6, 7syl2an 464 . . . . 5  |-  ( ( ( A  ~<  B  /\  B  ~<_  C )  /\  A  ~~  C )  ->  B  ~<_  A )
9 domnsym 7219 . . . . 5  |-  ( B  ~<_  A  ->  -.  A  ~<  B )
108, 9syl 16 . . . 4  |-  ( ( ( A  ~<  B  /\  B  ~<_  C )  /\  A  ~~  C )  ->  -.  A  ~<  B )
1110ex 424 . . 3  |-  ( ( A  ~<  B  /\  B  ~<_  C )  -> 
( A  ~~  C  ->  -.  A  ~<  B ) )
124, 11mt2d 111 . 2  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  -.  A  ~~  C )
13 brsdom 7116 . 2  |-  ( A 
~<  C  <->  ( A  ~<_  C  /\  -.  A  ~~  C ) )
143, 12, 13sylanbrc 646 1  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  A  ~<  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   class class class wbr 4199    ~~ cen 7092    ~<_ cdom 7093    ~< csdm 7094
This theorem is referenced by:  sdomentr  7227  sucdom  7290  sucdomiOLD  7291  infsdomnn  7354  fodomfib  7372  marypha1lem  7424  r1sdom  7684  infxpenlem  7879  infunsdom1  8077  fin56  8257  fodomb  8388  pwcfsdom  8442  cfpwsdom  8443  canthp1lem2  8512  gchhar  8530  gchpwdom  8533  gchina  8558  tsksdom  8615  tskpr  8629  tskcard  8640  gruina  8677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-er 6891  df-en 7096  df-dom 7097  df-sdom 7098
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