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Theorem pleval2i 15132
Description: One direction of pleval2 15133. (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 5714 . . . . . . . . 9  |-  ( X  e.  ( Base `  K
)  ->  K  e.  dom  Base )
2 pleval2.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
31, 2eleq2s 2533 . . . . . . . 8  |-  ( X  e.  B  ->  K  e.  dom  Base )
43adantr 465 . . . . . . 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 15128 . . . . . . . 8  |-  ( ( K  e.  dom  Base  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  X  =/= 
Y ) ) )
873expb 1188 . . . . . . 7  |-  ( ( K  e.  dom  Base  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  X  =/= 
Y ) ) )
94, 8mpancom 669 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  X  =/= 
Y ) ) )
109biimpar 485 . . . . 5  |-  ( ( ( X  e.  B  /\  Y  e.  B
)  /\  ( X  .<_  Y  /\  X  =/= 
Y ) )  ->  X  .<  Y )
1110expr 615 . . . 4  |-  ( ( ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  =/=  Y  ->  X  .<  Y ) )
1211necon1bd 2677 . . 3  |-  ( ( ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( -.  X  .<  Y  ->  X  =  Y ) )
1312orrd 378 . 2  |-  ( ( ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .<  Y  \/  X  =  Y ) )
1413ex 434 1  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
( X  .<  Y  \/  X  =  Y )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2604   class class class wbr 4290   dom cdm 4838   ` cfv 5416   Basecbs 14172   lecple 14243   ltcplt 15109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-iota 5379  df-fun 5418  df-fv 5424  df-plt 15126
This theorem is referenced by:  pleval2  15133  pospo  15141
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