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Theorem pleval2i 14114
Description: One direction of pleval2 14115. (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 5570 . . . . . . . . 9  |-  ( X  e.  ( Base `  K
)  ->  K  e.  dom  Base )
2 pleval2.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
31, 2eleq2s 2388 . . . . . . . 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 14110 . . . . . . . 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 2527 . . 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   dom cdm 4705   ` cfv 5271   Basecbs 13164   lecple 13231   ltcplt 14091
This theorem is referenced by:  pleval2  14115  pospo  14123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-plt 14108
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