Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lhpexle1lem Structured version   Unicode version

Theorem lhpexle1lem 35433
Description: Lemma for lhpexle1 35434 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 465 . . 3  |-  ( (
ph  /\  -.  X  e.  A )  ->  E. p  e.  A  ( p  .<_  W  /\  ps )
)
3 simprl 755 . . . . . 6  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  p  .<_  W )
4 simprr 756 . . . . . 6  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  ps )
5 simplr 754 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  p  e.  A )
6 simpllr 758 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  -.  X  e.  A )
7 nelne2 2771 . . . . . . 7  |-  ( ( p  e.  A  /\  -.  X  e.  A
)  ->  p  =/=  X )
85, 6, 7syl2anc 661 . . . . . 6  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  p  =/=  X )
93, 4, 83jca 1175 . . . . 5  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  (
p  .<_  W  /\  ps  /\  p  =/=  X ) )
109ex 434 . . . 4  |-  ( ( ( ph  /\  -.  X  e.  A )  /\  p  e.  A
)  ->  ( (
p  .<_  W  /\  ps )  ->  ( p  .<_  W  /\  ps  /\  p  =/=  X ) ) )
1110reximdva 2916 . . 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 465 . . 3  |-  ( (
ph  /\  -.  X  .<_  W )  ->  E. p  e.  A  ( p  .<_  W  /\  ps )
)
14 simprl 755 . . . . . 6  |-  ( ( ( ph  /\  -.  X  .<_  W )  /\  ( p  .<_  W  /\  ps ) )  ->  p  .<_  W )
15 simprr 756 . . . . . 6  |-  ( ( ( ph  /\  -.  X  .<_  W )  /\  ( p  .<_  W  /\  ps ) )  ->  ps )
16 simplr 754 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  .<_  W )  /\  ( p  .<_  W  /\  ps ) )  ->  -.  X  .<_  W )
17 nbrne2 4451 . . . . . . 7  |-  ( ( p  .<_  W  /\  -.  X  .<_  W )  ->  p  =/=  X
)
1814, 16, 17syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  -.  X  .<_  W )  /\  ( p  .<_  W  /\  ps ) )  ->  p  =/=  X )
1914, 15, 183jca 1175 . . . . 5  |-  ( ( ( ph  /\  -.  X  .<_  W )  /\  ( p  .<_  W  /\  ps ) )  ->  (
p  .<_  W  /\  ps  /\  p  =/=  X ) )
2019ex 434 . . . 4  |-  ( (
ph  /\  -.  X  .<_  W )  ->  (
( p  .<_  W  /\  ps )  ->  ( p 
.<_  W  /\  ps  /\  p  =/=  X ) ) )
2120reximdv 2915 . . 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 791 1  |-  ( ph  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 972    e. wcel 1802    =/= wne 2636   E.wrex 2792   class class class wbr 4433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-br 4434
This theorem is referenced by:  lhpexle1  35434  lhpexle2  35436  lhpexle3  35438
  Copyright terms: Public domain W3C validator