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Theorem sdomdomtr 6989
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 6884 . . 3  |-  ( A 
~<  B  ->  A  ~<_  B )
2 domtr 6909 . . 3  |-  ( ( A  ~<_  B  /\  B  ~<_  C )  ->  A  ~<_  C )
31, 2sylan 459 . 2  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  A  ~<_  C )
4 simpl 445 . . 3  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  A  ~<  B )
5 simpr 449 . . . . . 6  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  B  ~<_  C )
6 ensym 6905 . . . . . 6  |-  ( A 
~~  C  ->  C  ~~  A )
7 domentr 6915 . . . . . 6  |-  ( ( B  ~<_  C  /\  C  ~~  A )  ->  B  ~<_  A )
85, 6, 7syl2an 465 . . . . 5  |-  ( ( ( A  ~<  B  /\  B  ~<_  C )  /\  A  ~~  C )  ->  B  ~<_  A )
9 domnsym 6982 . . . . 5  |-  ( B  ~<_  A  ->  -.  A  ~<  B )
108, 9syl 17 . . . 4  |-  ( ( ( A  ~<  B  /\  B  ~<_  C )  /\  A  ~~  C )  ->  -.  A  ~<  B )
1110ex 425 . . 3  |-  ( ( A  ~<  B  /\  B  ~<_  C )  -> 
( A  ~~  C  ->  -.  A  ~<  B ) )
124, 11mt2d 111 . 2  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  -.  A  ~~  C )
13 brsdom 6879 . 2  |-  ( A 
~<  C  <->  ( A  ~<_  C  /\  -.  A  ~~  C ) )
143, 12, 13sylanbrc 648 1  |-  ( ( A  ~<  B  /\  B  ~<_  C )  ->  A  ~<  C )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360   class class class wbr 4024    ~~ cen 6855    ~<_ cdom 6856    ~< csdm 6857
This theorem is referenced by:  sdomentr  6990  sucdom  7053  sucdomiOLD  7054  infsdomnn  7113  fodomfib  7131  marypha1lem  7181  r1sdom  7441  infxpenlem  7636  infunsdom1  7834  fin56  8014  fodomb  8146  pwcfsdom  8200  cfpwsdom  8201  canthp1lem2  8270  gchhar  8288  gchpwdom  8291  gchina  8316  tsksdom  8373  tskpr  8387  tskcard  8398  gruina  8435
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861
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