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Theorem pleval2i 14421
Description: One direction of pleval2 14422. (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 5757 . . . . . . . . 9  |-  ( X  e.  ( Base `  K
)  ->  K  e.  dom  Base )
2 pleval2.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
31, 2eleq2s 2528 . . . . . . . 8  |-  ( X  e.  B  ->  K  e.  dom  Base )
43adantr 452 . . . . . . 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 14417 . . . . . . . 8  |-  ( ( K  e.  dom  Base  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  X  =/= 
Y ) ) )
873expb 1154 . . . . . . 7  |-  ( ( K  e.  dom  Base  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  X  =/= 
Y ) ) )
94, 8mpancom 651 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  X  =/= 
Y ) ) )
109biimpar 472 . . . . 5  |-  ( ( ( X  e.  B  /\  Y  e.  B
)  /\  ( X  .<_  Y  /\  X  =/= 
Y ) )  ->  X  .<  Y )
1110expr 599 . . . 4  |-  ( ( ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  =/=  Y  ->  X  .<  Y ) )
1211necon1bd 2672 . . 3  |-  ( ( ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( -.  X  .<  Y  ->  X  =  Y ) )
1312orrd 368 . 2  |-  ( ( ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .<  Y  \/  X  =  Y ) )
1413ex 424 1  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
( X  .<  Y  \/  X  =  Y )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   dom cdm 4878   ` cfv 5454   Basecbs 13469   lecple 13536   ltcplt 14398
This theorem is referenced by:  pleval2  14422  pospo  14430
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-plt 14415
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