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Theorem pmltpclem1 18804
Description: Lemma for pmltpc 18806. (Contributed by Mario Carneiro, 1-Jul-2014.)
Hypotheses
Ref Expression
pmltpclem1.1  |-  ( ph  ->  A  e.  S )
pmltpclem1.2  |-  ( ph  ->  B  e.  S )
pmltpclem1.3  |-  ( ph  ->  C  e.  S )
pmltpclem1.4  |-  ( ph  ->  A  <  B )
pmltpclem1.5  |-  ( ph  ->  B  <  C )
pmltpclem1.6  |-  ( ph  ->  ( ( ( F `
 A )  < 
( F `  B
)  /\  ( F `  C )  <  ( F `  B )
)  \/  ( ( F `  B )  <  ( F `  A )  /\  ( F `  B )  <  ( F `  C
) ) ) )
Assertion
Ref Expression
pmltpclem1  |-  ( ph  ->  E. a  e.  S  E. b  e.  S  E. c  e.  S  ( a  <  b  /\  b  <  c  /\  ( ( ( F `
 a )  < 
( F `  b
)  /\  ( F `  c )  <  ( F `  b )
)  \/  ( ( F `  b )  <  ( F `  a )  /\  ( F `  b )  <  ( F `  c
) ) ) ) )
Distinct variable groups:    a, b,
c, A    B, b,
c    C, c    F, a, b, c    S, a, b, c
Allowed substitution hints:    ph( a, b, c)    B( a)    C( a, b)

Proof of Theorem pmltpclem1
StepHypRef Expression
1 pmltpclem1.1 . 2  |-  ( ph  ->  A  e.  S )
2 pmltpclem1.2 . 2  |-  ( ph  ->  B  e.  S )
3 pmltpclem1.3 . 2  |-  ( ph  ->  C  e.  S )
4 pmltpclem1.4 . 2  |-  ( ph  ->  A  <  B )
5 pmltpclem1.5 . 2  |-  ( ph  ->  B  <  C )
6 pmltpclem1.6 . 2  |-  ( ph  ->  ( ( ( F `
 A )  < 
( F `  B
)  /\  ( F `  C )  <  ( F `  B )
)  \/  ( ( F `  B )  <  ( F `  A )  /\  ( F `  B )  <  ( F `  C
) ) ) )
7 breq1 4027 . . . 4  |-  ( a  =  A  ->  (
a  <  b  <->  A  <  b ) )
8 fveq2 5486 . . . . . . 7  |-  ( a  =  A  ->  ( F `  a )  =  ( F `  A ) )
98breq1d 4034 . . . . . 6  |-  ( a  =  A  ->  (
( F `  a
)  <  ( F `  b )  <->  ( F `  A )  <  ( F `  b )
) )
109anbi1d 685 . . . . 5  |-  ( a  =  A  ->  (
( ( F `  a )  <  ( F `  b )  /\  ( F `  c
)  <  ( F `  b ) )  <->  ( ( F `  A )  <  ( F `  b
)  /\  ( F `  c )  <  ( F `  b )
) ) )
118breq2d 4036 . . . . . 6  |-  ( a  =  A  ->  (
( F `  b
)  <  ( F `  a )  <->  ( F `  b )  <  ( F `  A )
) )
1211anbi1d 685 . . . . 5  |-  ( a  =  A  ->  (
( ( F `  b )  <  ( F `  a )  /\  ( F `  b
)  <  ( F `  c ) )  <->  ( ( F `  b )  <  ( F `  A
)  /\  ( F `  b )  <  ( F `  c )
) ) )
1310, 12orbi12d 690 . . . 4  |-  ( a  =  A  ->  (
( ( ( F `
 a )  < 
( F `  b
)  /\  ( F `  c )  <  ( F `  b )
)  \/  ( ( F `  b )  <  ( F `  a )  /\  ( F `  b )  <  ( F `  c
) ) )  <->  ( (
( F `  A
)  <  ( F `  b )  /\  ( F `  c )  <  ( F `  b
) )  \/  (
( F `  b
)  <  ( F `  A )  /\  ( F `  b )  <  ( F `  c
) ) ) ) )
147, 133anbi13d 1254 . . 3  |-  ( a  =  A  ->  (
( a  <  b  /\  b  <  c  /\  ( ( ( F `
 a )  < 
( F `  b
)  /\  ( F `  c )  <  ( F `  b )
)  \/  ( ( F `  b )  <  ( F `  a )  /\  ( F `  b )  <  ( F `  c
) ) ) )  <-> 
( A  <  b  /\  b  <  c  /\  ( ( ( F `
 A )  < 
( F `  b
)  /\  ( F `  c )  <  ( F `  b )
)  \/  ( ( F `  b )  <  ( F `  A )  /\  ( F `  b )  <  ( F `  c
) ) ) ) ) )
15 breq2 4028 . . . 4  |-  ( b  =  B  ->  ( A  <  b  <->  A  <  B ) )
16 breq1 4027 . . . 4  |-  ( b  =  B  ->  (
b  <  c  <->  B  <  c ) )
17 fveq2 5486 . . . . . . 7  |-  ( b  =  B  ->  ( F `  b )  =  ( F `  B ) )
1817breq2d 4036 . . . . . 6  |-  ( b  =  B  ->  (
( F `  A
)  <  ( F `  b )  <->  ( F `  A )  <  ( F `  B )
) )
1917breq2d 4036 . . . . . 6  |-  ( b  =  B  ->  (
( F `  c
)  <  ( F `  b )  <->  ( F `  c )  <  ( F `  B )
) )
2018, 19anbi12d 691 . . . . 5  |-  ( b  =  B  ->  (
( ( F `  A )  <  ( F `  b )  /\  ( F `  c
)  <  ( F `  b ) )  <->  ( ( F `  A )  <  ( F `  B
)  /\  ( F `  c )  <  ( F `  B )
) ) )
2117breq1d 4034 . . . . . 6  |-  ( b  =  B  ->  (
( F `  b
)  <  ( F `  A )  <->  ( F `  B )  <  ( F `  A )
) )
2217breq1d 4034 . . . . . 6  |-  ( b  =  B  ->  (
( F `  b
)  <  ( F `  c )  <->  ( F `  B )  <  ( F `  c )
) )
2321, 22anbi12d 691 . . . . 5  |-  ( b  =  B  ->  (
( ( F `  b )  <  ( F `  A )  /\  ( F `  b
)  <  ( F `  c ) )  <->  ( ( F `  B )  <  ( F `  A
)  /\  ( F `  B )  <  ( F `  c )
) ) )
2420, 23orbi12d 690 . . . 4  |-  ( b  =  B  ->  (
( ( ( F `
 A )  < 
( F `  b
)  /\  ( F `  c )  <  ( F `  b )
)  \/  ( ( F `  b )  <  ( F `  A )  /\  ( F `  b )  <  ( F `  c
) ) )  <->  ( (
( F `  A
)  <  ( F `  B )  /\  ( F `  c )  <  ( F `  B
) )  \/  (
( F `  B
)  <  ( F `  A )  /\  ( F `  B )  <  ( F `  c
) ) ) ) )
2515, 16, 243anbi123d 1252 . . 3  |-  ( b  =  B  ->  (
( A  <  b  /\  b  <  c  /\  ( ( ( F `
 A )  < 
( F `  b
)  /\  ( F `  c )  <  ( F `  b )
)  \/  ( ( F `  b )  <  ( F `  A )  /\  ( F `  b )  <  ( F `  c
) ) ) )  <-> 
( A  <  B  /\  B  <  c  /\  ( ( ( F `
 A )  < 
( F `  B
)  /\  ( F `  c )  <  ( F `  B )
)  \/  ( ( F `  B )  <  ( F `  A )  /\  ( F `  B )  <  ( F `  c
) ) ) ) ) )
26 breq2 4028 . . . 4  |-  ( c  =  C  ->  ( B  <  c  <->  B  <  C ) )
27 fveq2 5486 . . . . . . 7  |-  ( c  =  C  ->  ( F `  c )  =  ( F `  C ) )
2827breq1d 4034 . . . . . 6  |-  ( c  =  C  ->  (
( F `  c
)  <  ( F `  B )  <->  ( F `  C )  <  ( F `  B )
) )
2928anbi2d 684 . . . . 5  |-  ( c  =  C  ->  (
( ( F `  A )  <  ( F `  B )  /\  ( F `  c
)  <  ( F `  B ) )  <->  ( ( F `  A )  <  ( F `  B
)  /\  ( F `  C )  <  ( F `  B )
) ) )
3027breq2d 4036 . . . . . 6  |-  ( c  =  C  ->  (
( F `  B
)  <  ( F `  c )  <->  ( F `  B )  <  ( F `  C )
) )
3130anbi2d 684 . . . . 5  |-  ( c  =  C  ->  (
( ( F `  B )  <  ( F `  A )  /\  ( F `  B
)  <  ( F `  c ) )  <->  ( ( F `  B )  <  ( F `  A
)  /\  ( F `  B )  <  ( F `  C )
) ) )
3229, 31orbi12d 690 . . . 4  |-  ( c  =  C  ->  (
( ( ( F `
 A )  < 
( F `  B
)  /\  ( F `  c )  <  ( F `  B )
)  \/  ( ( F `  B )  <  ( F `  A )  /\  ( F `  B )  <  ( F `  c
) ) )  <->  ( (
( F `  A
)  <  ( F `  B )  /\  ( F `  C )  <  ( F `  B
) )  \/  (
( F `  B
)  <  ( F `  A )  /\  ( F `  B )  <  ( F `  C
) ) ) ) )
3326, 323anbi23d 1255 . . 3  |-  ( c  =  C  ->  (
( A  <  B  /\  B  <  c  /\  ( ( ( F `
 A )  < 
( F `  B
)  /\  ( F `  c )  <  ( F `  B )
)  \/  ( ( F `  B )  <  ( F `  A )  /\  ( F `  B )  <  ( F `  c
) ) ) )  <-> 
( A  <  B  /\  B  <  C  /\  ( ( ( F `
 A )  < 
( F `  B
)  /\  ( F `  C )  <  ( F `  B )
)  \/  ( ( F `  B )  <  ( F `  A )  /\  ( F `  B )  <  ( F `  C
) ) ) ) ) )
3414, 25, 33rspc3ev 2895 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  /\  ( A  <  B  /\  B  < 
C  /\  ( (
( F `  A
)  <  ( F `  B )  /\  ( F `  C )  <  ( F `  B
) )  \/  (
( F `  B
)  <  ( F `  A )  /\  ( F `  B )  <  ( F `  C
) ) ) ) )  ->  E. a  e.  S  E. b  e.  S  E. c  e.  S  ( a  <  b  /\  b  < 
c  /\  ( (
( F `  a
)  <  ( F `  b )  /\  ( F `  c )  <  ( F `  b
) )  \/  (
( F `  b
)  <  ( F `  a )  /\  ( F `  b )  <  ( F `  c
) ) ) ) )
351, 2, 3, 4, 5, 6, 34syl33anc 1197 1  |-  ( ph  ->  E. a  e.  S  E. b  e.  S  E. c  e.  S  ( a  <  b  /\  b  <  c  /\  ( ( ( F `
 a )  < 
( F `  b
)  /\  ( F `  c )  <  ( F `  b )
)  \/  ( ( F `  b )  <  ( F `  a )  /\  ( F `  b )  <  ( F `  c
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1685   E.wrex 2545   class class class wbr 4024   ` cfv 5221    < clt 8863
This theorem is referenced by:  pmltpclem2  18805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229
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