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Theorem pleval2i 14098
Description: One direction of pleval2 14099. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
pleval2.b  |-  B  =  ( Base `  K
)
pleval2.l  |-  .<_  =  ( le `  K )
pleval2.s  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
pleval2i  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
( X  .<  Y  \/  X  =  Y )
) )

Proof of Theorem pleval2i
StepHypRef Expression
1 elfvdm 5554 . . . . . . . . 9  |-  ( X  e.  ( Base `  K
)  ->  K  e.  dom  Base )
2 pleval2.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
31, 2eleq2s 2375 . . . . . . . 8  |-  ( X  e.  B  ->  K  e.  dom  Base )
43adantr 451 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  K  e.  dom  Base )
5 pleval2.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
6 pleval2.s . . . . . . . . 9  |-  .<  =  ( lt `  K )
75, 6pltval 14094 . . . . . . . 8  |-  ( ( K  e.  dom  Base  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  X  =/= 
Y ) ) )
873expb 1152 . . . . . . 7  |-  ( ( K  e.  dom  Base  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  X  =/= 
Y ) ) )
94, 8mpancom 650 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  X  =/= 
Y ) ) )
109biimpar 471 . . . . 5  |-  ( ( ( X  e.  B  /\  Y  e.  B
)  /\  ( X  .<_  Y  /\  X  =/= 
Y ) )  ->  X  .<  Y )
1110expr 598 . . . 4  |-  ( ( ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  =/=  Y  ->  X  .<  Y ) )
1211necon1bd 2514 . . 3  |-  ( ( ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( -.  X  .<  Y  ->  X  =  Y ) )
1312orrd 367 . 2  |-  ( ( ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .<  Y  \/  X  =  Y ) )
1413ex 423 1  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
( X  .<  Y  \/  X  =  Y )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   dom cdm 4689   ` cfv 5255   Basecbs 13148   lecple 13215   ltcplt 14075
This theorem is referenced by:  pleval2  14099  pospo  14107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-plt 14092
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