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Theorem lhpexle1lem 30741
Description: Lemma for lhpexle1 30742 and others that eliminates restrictions on  X. (Contributed by NM, 24-Jul-2013.)
Hypotheses
Ref Expression
lhpexle1lem.1  |-  ( ph  ->  E. p  e.  A  ( p  .<_  W  /\  ps ) )
lhpexle1lem.2  |-  ( (
ph  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X ) )
Assertion
Ref Expression
lhpexle1lem  |-  ( ph  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X
) )
Distinct variable groups:    .<_ , p    A, p    W, p    X, p    ph, p
Allowed substitution hint:    ps( p)

Proof of Theorem lhpexle1lem
StepHypRef Expression
1 lhpexle1lem.1 . . . 4  |-  ( ph  ->  E. p  e.  A  ( p  .<_  W  /\  ps ) )
21adantr 452 . . 3  |-  ( (
ph  /\  -.  X  e.  A )  ->  E. p  e.  A  ( p  .<_  W  /\  ps )
)
3 simprl 733 . . . . . 6  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  p  .<_  W )
4 simprr 734 . . . . . 6  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  ps )
5 simplr 732 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  p  e.  A )
6 simpllr 736 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  -.  X  e.  A )
7 nelne2 2688 . . . . . . 7  |-  ( ( p  e.  A  /\  -.  X  e.  A
)  ->  p  =/=  X )
85, 6, 7syl2anc 643 . . . . . 6  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  p  =/=  X )
93, 4, 83jca 1134 . . . . 5  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  (
p  .<_  W  /\  ps  /\  p  =/=  X ) )
109ex 424 . . . 4  |-  ( ( ( ph  /\  -.  X  e.  A )  /\  p  e.  A
)  ->  ( (
p  .<_  W  /\  ps )  ->  ( p  .<_  W  /\  ps  /\  p  =/=  X ) ) )
1110reximdva 2810 . . 3  |-  ( (
ph  /\  -.  X  e.  A )  ->  ( E. p  e.  A  ( p  .<_  W  /\  ps )  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X ) ) )
122, 11mpd 15 . 2  |-  ( (
ph  /\  -.  X  e.  A )  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X ) )
131adantr 452 . . 3  |-  ( (
ph  /\  -.  X  .<_  W )  ->  E. p  e.  A  ( p  .<_  W  /\  ps )
)
14 simprl 733 . . . . . 6  |-  ( ( ( ph  /\  -.  X  .<_  W )  /\  ( p  .<_  W  /\  ps ) )  ->  p  .<_  W )
15 simprr 734 . . . . . 6  |-  ( ( ( ph  /\  -.  X  .<_  W )  /\  ( p  .<_  W  /\  ps ) )  ->  ps )
16 simplr 732 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  .<_  W )  /\  ( p  .<_  W  /\  ps ) )  ->  -.  X  .<_  W )
17 nbrne2 4222 . . . . . . 7  |-  ( ( p  .<_  W  /\  -.  X  .<_  W )  ->  p  =/=  X
)
1814, 16, 17syl2anc 643 . . . . . 6  |-  ( ( ( ph  /\  -.  X  .<_  W )  /\  ( p  .<_  W  /\  ps ) )  ->  p  =/=  X )
1914, 15, 183jca 1134 . . . . 5  |-  ( ( ( ph  /\  -.  X  .<_  W )  /\  ( p  .<_  W  /\  ps ) )  ->  (
p  .<_  W  /\  ps  /\  p  =/=  X ) )
2019ex 424 . . . 4  |-  ( (
ph  /\  -.  X  .<_  W )  ->  (
( p  .<_  W  /\  ps )  ->  ( p 
.<_  W  /\  ps  /\  p  =/=  X ) ) )
2120reximdv 2809 . . 3  |-  ( (
ph  /\  -.  X  .<_  W )  ->  ( E. p  e.  A  ( p  .<_  W  /\  ps )  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X ) ) )
2213, 21mpd 15 . 2  |-  ( (
ph  /\  -.  X  .<_  W )  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X ) )
23 lhpexle1lem.2 . 2  |-  ( (
ph  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X ) )
2412, 22, 23pm2.61dda 769 1  |-  ( ph  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725    =/= wne 2598   E.wrex 2698   class class class wbr 4204
This theorem is referenced by:  lhpexle1  30742  lhpexle2  30744  lhpexle3  30746
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205
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