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Theorem ltexprlemdisj 7596
Description: Our constructed difference is disjoint. Lemma for ltexpri 7603. (Contributed by Jim Kingdon, 17-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
Assertion
Ref Expression
ltexprlemdisj  |-  ( A 
<P  B  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  C
)  /\  q  e.  ( 2nd `  C ) ) )
Distinct variable groups:    x, y, q, A    x, B, y, q    x, C, y, q

Proof of Theorem ltexprlemdisj
Dummy variables  z  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltsonq 7388 . . . . . 6  |-  <Q  Or  Q.
2 ltrelnq 7355 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
31, 2son2lpi 5021 . . . . 5  |-  -.  (
y  <Q  z  /\  z  <Q  y )
4 ltrelpr 7495 . . . . . . . . . . . . . . . 16  |-  <P  C_  ( P.  X.  P. )
54brel 4675 . . . . . . . . . . . . . . 15  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
65simprd 114 . . . . . . . . . . . . . 14  |-  ( A 
<P  B  ->  B  e. 
P. )
7 prop 7465 . . . . . . . . . . . . . 14  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
86, 7syl 14 . . . . . . . . . . . . 13  |-  ( A 
<P  B  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
9 prltlu 7477 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  ( y  +Q  q
)  e.  ( 1st `  B )  /\  (
z  +Q  q )  e.  ( 2nd `  B
) )  ->  (
y  +Q  q ) 
<Q  ( z  +Q  q
) )
108, 9syl3an1 1271 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( y  +Q  q
)  e.  ( 1st `  B )  /\  (
z  +Q  q )  e.  ( 2nd `  B
) )  ->  (
y  +Q  q ) 
<Q  ( z  +Q  q
) )
11103expb 1204 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( ( y  +Q  q )  e.  ( 1st `  B )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  ->  (
y  +Q  q ) 
<Q  ( z  +Q  q
) )
1211adantlr 477 . . . . . . . . . 10  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  +Q  q )  e.  ( 1st `  B )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  ->  (
y  +Q  q ) 
<Q  ( z  +Q  q
) )
1312adantrll 484 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  +Q  q )  e.  ( 2nd `  B
) ) )  -> 
( y  +Q  q
)  <Q  ( z  +Q  q ) )
1413adantrrl 486 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
( y  +Q  q
)  <Q  ( z  +Q  q ) )
15 ltanqg 7390 . . . . . . . . . 10  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
1615adantl 277 . . . . . . . . 9  |-  ( ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
175simpld 112 . . . . . . . . . . . . 13  |-  ( A 
<P  B  ->  A  e. 
P. )
18 prop 7465 . . . . . . . . . . . . 13  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
1917, 18syl 14 . . . . . . . . . . . 12  |-  ( A 
<P  B  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
20 elprnqu 7472 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2119, 20sylan 283 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2221ad2ant2r 509 . . . . . . . . . 10  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  ->  y  e.  Q. )
2322adantrr 479 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
y  e.  Q. )
24 elprnql 7471 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
2519, 24sylan 283 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
2625ad2ant2r 509 . . . . . . . . . 10  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( z  e.  ( 1st `  A )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  ->  z  e.  Q. )
2726adantrl 478 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
z  e.  Q. )
28 simplr 528 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
q  e.  Q. )
29 addcomnqg 7371 . . . . . . . . . 10  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
3029adantl 277 . . . . . . . . 9  |-  ( ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  /\  ( f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  =  ( g  +Q  f ) )
3116, 23, 27, 28, 30caovord2d 6038 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
( y  <Q  z  <->  ( y  +Q  q ) 
<Q  ( z  +Q  q
) ) )
3214, 31mpbird 167 . . . . . . 7  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
y  <Q  z )
33 prltlu 7477 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A
) )  ->  z  <Q  y )
3419, 33syl3an1 1271 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A
) )  ->  z  <Q  y )
35343com23 1209 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  y  e.  ( 2nd `  A )  /\  z  e.  ( 1st `  A
) )  ->  z  <Q  y )
36353expb 1204 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  ( y  e.  ( 2nd `  A )  /\  z  e.  ( 1st `  A ) ) )  ->  z  <Q  y )
3736adantlr 477 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( y  e.  ( 2nd `  A )  /\  z  e.  ( 1st `  A ) ) )  ->  z  <Q  y )
3837adantrlr 485 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  z  e.  ( 1st `  A
) ) )  -> 
z  <Q  y )
3938adantrrr 487 . . . . . . 7  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
z  <Q  y )
4032, 39jca 306 . . . . . 6  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
( y  <Q  z  /\  z  <Q  y ) )
4140ex 115 . . . . 5  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  ( ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  ->  (
y  <Q  z  /\  z  <Q  y ) ) )
423, 41mtoi 664 . . . 4  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  -.  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
4342alrimivv 1875 . . 3  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  A. y A. z  -.  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
44 ltexprlem.1 . . . . . . . . . . . 12  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
4544ltexprlemell 7588 . . . . . . . . . . 11  |-  ( q  e.  ( 1st `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
4644ltexprlemelu 7589 . . . . . . . . . . 11  |-  ( q  e.  ( 2nd `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )
4745, 46anbi12i 460 . . . . . . . . . 10  |-  ( ( q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C
) )  <->  ( (
q  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  /\  (
q  e.  Q.  /\  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
48 anandi 590 . . . . . . . . . 10  |-  ( ( q  e.  Q.  /\  ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )  <->  ( (
q  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  /\  (
q  e.  Q.  /\  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
4947, 48bitr4i 187 . . . . . . . . 9  |-  ( ( q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C
) )  <->  ( q  e.  Q.  /\  ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
5049baib 919 . . . . . . . 8  |-  ( q  e.  Q.  ->  (
( q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C ) )  <->  ( E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
51 eleq1 2240 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
y  e.  ( 1st `  A )  <->  z  e.  ( 1st `  A ) ) )
52 oveq1 5876 . . . . . . . . . . . 12  |-  ( y  =  z  ->  (
y  +Q  q )  =  ( z  +Q  q ) )
5352eleq1d 2246 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
( y  +Q  q
)  e.  ( 2nd `  B )  <->  ( z  +Q  q )  e.  ( 2nd `  B ) ) )
5451, 53anbi12d 473 . . . . . . . . . 10  |-  ( y  =  z  ->  (
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) )  <->  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
5554cbvexv 1918 . . . . . . . . 9  |-  ( E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) )  <->  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )
5655anbi2i 457 . . . . . . . 8  |-  ( ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) )  <->  ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
5750, 56bitrdi 196 . . . . . . 7  |-  ( q  e.  Q.  ->  (
( q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C ) )  <->  ( E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
58 eeanv 1932 . . . . . . 7  |-  ( E. y E. z ( ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  <->  ( E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  E. z
( z  e.  ( 1st `  A )  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
5957, 58bitr4di 198 . . . . . 6  |-  ( q  e.  Q.  ->  (
( q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C ) )  <->  E. y E. z
( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
6059notbid 667 . . . . 5  |-  ( q  e.  Q.  ->  ( -.  ( q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C ) )  <->  -.  E. y E. z ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
61 alnex 1499 . . . . . . 7  |-  ( A. z  -.  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  <->  -.  E. z
( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
6261albii 1470 . . . . . 6  |-  ( A. y A. z  -.  (
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  <->  A. y  -.  E. z ( ( y  e.  ( 2nd `  A )  /\  (
y  +Q  q )  e.  ( 1st `  B
) )  /\  (
z  e.  ( 1st `  A )  /\  (
z  +Q  q )  e.  ( 2nd `  B
) ) ) )
63 alnex 1499 . . . . . 6  |-  ( A. y  -.  E. z ( ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  <->  -.  E. y E. z ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
6462, 63bitri 184 . . . . 5  |-  ( A. y A. z  -.  (
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) )  <->  -.  E. y E. z ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) )
6560, 64bitr4di 198 . . . 4  |-  ( q  e.  Q.  ->  ( -.  ( q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C ) )  <->  A. y A. z  -.  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
6665adantl 277 . . 3  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  ( -.  ( q  e.  ( 1st `  C
)  /\  q  e.  ( 2nd `  C ) )  <->  A. y A. z  -.  ( ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) )  /\  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
6743, 66mpbird 167 . 2  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  C
)  /\  q  e.  ( 2nd `  C ) ) )
6867ralrimiva 2550 1  |-  ( A 
<P  B  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  C
)  /\  q  e.  ( 2nd `  C ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978   A.wal 1351    = wceq 1353   E.wex 1492    e. wcel 2148   A.wral 2455   {crab 2459   <.cop 3594   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   Q.cnq 7270    +Q cplq 7272    <Q cltq 7275   P.cnp 7281    <P cltp 7285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-ltnqqs 7343  df-inp 7456  df-iltp 7460
This theorem is referenced by:  ltexprlempr  7598
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