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Theorem pleval2i 15444
Description: One direction of pleval2 15445. (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 5890 . . . . . . . . 9  |-  ( X  e.  ( Base `  K
)  ->  K  e.  dom  Base )
2 pleval2.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
31, 2eleq2s 2575 . . . . . . . 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 15440 . . . . . . . 8  |-  ( ( K  e.  dom  Base  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  X  =/= 
Y ) ) )
873expb 1197 . . . . . . 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 2685 . . 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 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   dom cdm 4999   ` cfv 5586   Basecbs 14483   lecple 14555   ltcplt 15421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-plt 15438
This theorem is referenced by:  pleval2  15445  pospo  15453
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