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Theorem ltsopr 6752
Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4062). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
Assertion
Ref Expression
ltsopr  |-  <P  Or  P.

Proof of Theorem ltsopr
Dummy variables  r  q  s  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltpopr 6751 . 2  |-  <P  Po  P.
2 ltdfpr 6662 . . . . 5  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  <P  y  <->  E. q  e.  Q.  (
q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y
) ) ) )
323adant3 935 . . . 4  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  ->  (
x  <P  y  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  x
)  /\  q  e.  ( 1st `  y ) ) ) )
4 prop 6631 . . . . . . . . . . . 12  |-  ( x  e.  P.  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  P. )
5 prnminu 6645 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  P.  /\  q  e.  ( 2nd `  x ) )  ->  E. r  e.  ( 2nd `  x ) r 
<Q  q )
64, 5sylan 271 . . . . . . . . . . 11  |-  ( ( x  e.  P.  /\  q  e.  ( 2nd `  x ) )  ->  E. r  e.  ( 2nd `  x ) r 
<Q  q )
7 prop 6631 . . . . . . . . . . . 12  |-  ( y  e.  P.  ->  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  e.  P. )
8 prnmaxl 6644 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  P.  /\  q  e.  ( 1st `  y ) )  ->  E. s  e.  ( 1st `  y ) q 
<Q  s )
97, 8sylan 271 . . . . . . . . . . 11  |-  ( ( y  e.  P.  /\  q  e.  ( 1st `  y ) )  ->  E. s  e.  ( 1st `  y ) q 
<Q  s )
106, 9anim12i 325 . . . . . . . . . 10  |-  ( ( ( x  e.  P.  /\  q  e.  ( 2nd `  x ) )  /\  ( y  e.  P.  /\  q  e.  ( 1st `  y ) ) )  ->  ( E. r  e.  ( 2nd `  x
) r  <Q  q  /\  E. s  e.  ( 1st `  y ) q  <Q  s )
)
1110an4s 530 . . . . . . . . 9  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) )  ->  ( E. r  e.  ( 2nd `  x ) r 
<Q  q  /\  E. s  e.  ( 1st `  y
) q  <Q  s
) )
12 reeanv 2496 . . . . . . . . 9  |-  ( E. r  e.  ( 2nd `  x ) E. s  e.  ( 1st `  y
) ( r  <Q 
q  /\  q  <Q  s )  <->  ( E. r  e.  ( 2nd `  x
) r  <Q  q  /\  E. s  e.  ( 1st `  y ) q  <Q  s )
)
1311, 12sylibr 141 . . . . . . . 8  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) )  ->  E. r  e.  ( 2nd `  x
) E. s  e.  ( 1st `  y
) ( r  <Q 
q  /\  q  <Q  s ) )
14133adantl3 1073 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  /\  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) )  ->  E. r  e.  ( 2nd `  x
) E. s  e.  ( 1st `  y
) ( r  <Q 
q  /\  q  <Q  s ) )
15 ltsonq 6554 . . . . . . . . . . . . 13  |-  <Q  Or  Q.
16 ltrelnq 6521 . . . . . . . . . . . . 13  |-  <Q  C_  ( Q.  X.  Q. )
1715, 16sotri 4748 . . . . . . . . . . . 12  |-  ( ( r  <Q  q  /\  q  <Q  s )  -> 
r  <Q  s )
1817adantl 266 . . . . . . . . . . 11  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e. 
P. )  /\  (
q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y
) ) )  /\  ( r  e.  ( 2nd `  x )  /\  s  e.  ( 1st `  y ) ) )  /\  (
r  <Q  q  /\  q  <Q  s ) )  -> 
r  <Q  s )
19 prop 6631 . . . . . . . . . . . . . . . 16  |-  ( z  e.  P.  ->  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  e.  P. )
20 prloc 6647 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  z
) ,  ( 2nd `  z ) >.  e.  P.  /\  r  <Q  s )  ->  ( r  e.  ( 1st `  z )  \/  s  e.  ( 2nd `  z ) ) )
2119, 20sylan 271 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  P.  /\  r  <Q  s )  -> 
( r  e.  ( 1st `  z )  \/  s  e.  ( 2nd `  z ) ) )
22213ad2antl3 1079 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  /\  r  <Q  s )  ->  ( r  e.  ( 1st `  z
)  \/  s  e.  ( 2nd `  z
) ) )
2322ex 112 . . . . . . . . . . . . 13  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  ->  (
r  <Q  s  ->  (
r  e.  ( 1st `  z )  \/  s  e.  ( 2nd `  z
) ) ) )
2423adantr 265 . . . . . . . . . . . 12  |-  ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  /\  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) )  ->  (
r  <Q  s  ->  (
r  e.  ( 1st `  z )  \/  s  e.  ( 2nd `  z
) ) ) )
2524ad2antrr 465 . . . . . . . . . . 11  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e. 
P. )  /\  (
q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y
) ) )  /\  ( r  e.  ( 2nd `  x )  /\  s  e.  ( 1st `  y ) ) )  /\  (
r  <Q  q  /\  q  <Q  s ) )  -> 
( r  <Q  s  ->  ( r  e.  ( 1st `  z )  \/  s  e.  ( 2nd `  z ) ) ) )
2618, 25mpd 13 . . . . . . . . . 10  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e. 
P. )  /\  (
q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y
) ) )  /\  ( r  e.  ( 2nd `  x )  /\  s  e.  ( 1st `  y ) ) )  /\  (
r  <Q  q  /\  q  <Q  s ) )  -> 
( r  e.  ( 1st `  z )  \/  s  e.  ( 2nd `  z ) ) )
27 elprnqu 6638 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  P.  /\  r  e.  ( 2nd `  x ) )  -> 
r  e.  Q. )
284, 27sylan 271 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  P.  /\  r  e.  ( 2nd `  x ) )  -> 
r  e.  Q. )
29 ax-ia3 105 . . . . . . . . . . . . . . . . . . . . 21  |-  ( r  e.  ( 2nd `  x
)  ->  ( r  e.  ( 1st `  z
)  ->  ( r  e.  ( 2nd `  x
)  /\  r  e.  ( 1st `  z ) ) ) )
3029adantl 266 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  P.  /\  r  e.  ( 2nd `  x ) )  -> 
( r  e.  ( 1st `  z )  ->  ( r  e.  ( 2nd `  x
)  /\  r  e.  ( 1st `  z ) ) ) )
31 19.8a 1498 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( r  e.  Q.  /\  ( r  e.  ( 2nd `  x )  /\  r  e.  ( 1st `  z ) ) )  ->  E. r
( r  e.  Q.  /\  ( r  e.  ( 2nd `  x )  /\  r  e.  ( 1st `  z ) ) ) )
3228, 30, 31syl6an 1339 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  P.  /\  r  e.  ( 2nd `  x ) )  -> 
( r  e.  ( 1st `  z )  ->  E. r ( r  e.  Q.  /\  (
r  e.  ( 2nd `  x )  /\  r  e.  ( 1st `  z
) ) ) ) )
33323ad2antl1 1077 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  /\  r  e.  ( 2nd `  x ) )  ->  ( r  e.  ( 1st `  z
)  ->  E. r
( r  e.  Q.  /\  ( r  e.  ( 2nd `  x )  /\  r  e.  ( 1st `  z ) ) ) ) )
3433imp 119 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P.  /\  z  e.  P. )  /\  r  e.  ( 2nd `  x ) )  /\  r  e.  ( 1st `  z
) )  ->  E. r
( r  e.  Q.  /\  ( r  e.  ( 2nd `  x )  /\  r  e.  ( 1st `  z ) ) ) )
35 df-rex 2329 . . . . . . . . . . . . . . . . 17  |-  ( E. r  e.  Q.  (
r  e.  ( 2nd `  x )  /\  r  e.  ( 1st `  z
) )  <->  E. r
( r  e.  Q.  /\  ( r  e.  ( 2nd `  x )  /\  r  e.  ( 1st `  z ) ) ) )
3634, 35sylibr 141 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P.  /\  z  e.  P. )  /\  r  e.  ( 2nd `  x ) )  /\  r  e.  ( 1st `  z
) )  ->  E. r  e.  Q.  ( r  e.  ( 2nd `  x
)  /\  r  e.  ( 1st `  z ) ) )
37 ltdfpr 6662 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  <P  z  <->  E. r  e.  Q.  (
r  e.  ( 2nd `  x )  /\  r  e.  ( 1st `  z
) ) ) )
3837biimprd 151 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( E. r  e. 
Q.  ( r  e.  ( 2nd `  x
)  /\  r  e.  ( 1st `  z ) )  ->  x  <P  z ) )
39383adant2 934 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  ->  ( E. r  e.  Q.  ( r  e.  ( 2nd `  x )  /\  r  e.  ( 1st `  z ) )  ->  x  <P  z ) )
4039ad2antrr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P.  /\  z  e.  P. )  /\  r  e.  ( 2nd `  x ) )  /\  r  e.  ( 1st `  z
) )  ->  ( E. r  e.  Q.  ( r  e.  ( 2nd `  x )  /\  r  e.  ( 1st `  z ) )  ->  x  <P  z ) )
4136, 40mpd 13 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P.  /\  z  e.  P. )  /\  r  e.  ( 2nd `  x ) )  /\  r  e.  ( 1st `  z
) )  ->  x  <P  z )
4241ex 112 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  /\  r  e.  ( 2nd `  x ) )  ->  ( r  e.  ( 1st `  z
)  ->  x  <P  z ) )
4342adantrr 456 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  /\  ( r  e.  ( 2nd `  x )  /\  s  e.  ( 1st `  y ) ) )  ->  (
r  e.  ( 1st `  z )  ->  x  <P  z ) )
44 elprnql 6637 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  P.  /\  s  e.  ( 1st `  y ) )  -> 
s  e.  Q. )
457, 44sylan 271 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  P.  /\  s  e.  ( 1st `  y ) )  -> 
s  e.  Q. )
46 pm3.21 255 . . . . . . . . . . . . . . . . . . . . 21  |-  ( s  e.  ( 1st `  y
)  ->  ( s  e.  ( 2nd `  z
)  ->  ( s  e.  ( 2nd `  z
)  /\  s  e.  ( 1st `  y ) ) ) )
4746adantl 266 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  P.  /\  s  e.  ( 1st `  y ) )  -> 
( s  e.  ( 2nd `  z )  ->  ( s  e.  ( 2nd `  z
)  /\  s  e.  ( 1st `  y ) ) ) )
48 19.8a 1498 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( s  e.  Q.  /\  ( s  e.  ( 2nd `  z )  /\  s  e.  ( 1st `  y ) ) )  ->  E. s
( s  e.  Q.  /\  ( s  e.  ( 2nd `  z )  /\  s  e.  ( 1st `  y ) ) ) )
4945, 47, 48syl6an 1339 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  P.  /\  s  e.  ( 1st `  y ) )  -> 
( s  e.  ( 2nd `  z )  ->  E. s ( s  e.  Q.  /\  (
s  e.  ( 2nd `  z )  /\  s  e.  ( 1st `  y
) ) ) ) )
50493ad2antl2 1078 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  /\  s  e.  ( 1st `  y ) )  ->  ( s  e.  ( 2nd `  z
)  ->  E. s
( s  e.  Q.  /\  ( s  e.  ( 2nd `  z )  /\  s  e.  ( 1st `  y ) ) ) ) )
5150imp 119 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P.  /\  z  e.  P. )  /\  s  e.  ( 1st `  y ) )  /\  s  e.  ( 2nd `  z
) )  ->  E. s
( s  e.  Q.  /\  ( s  e.  ( 2nd `  z )  /\  s  e.  ( 1st `  y ) ) ) )
52 df-rex 2329 . . . . . . . . . . . . . . . . 17  |-  ( E. s  e.  Q.  (
s  e.  ( 2nd `  z )  /\  s  e.  ( 1st `  y
) )  <->  E. s
( s  e.  Q.  /\  ( s  e.  ( 2nd `  z )  /\  s  e.  ( 1st `  y ) ) ) )
5351, 52sylibr 141 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P.  /\  z  e.  P. )  /\  s  e.  ( 1st `  y ) )  /\  s  e.  ( 2nd `  z
) )  ->  E. s  e.  Q.  ( s  e.  ( 2nd `  z
)  /\  s  e.  ( 1st `  y ) ) )
54 ltdfpr 6662 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  e.  P.  /\  y  e.  P. )  ->  ( z  <P  y  <->  E. s  e.  Q.  (
s  e.  ( 2nd `  z )  /\  s  e.  ( 1st `  y
) ) ) )
5554biimprd 151 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  e.  P.  /\  y  e.  P. )  ->  ( E. s  e. 
Q.  ( s  e.  ( 2nd `  z
)  /\  s  e.  ( 1st `  y ) )  ->  z  <P  y ) )
5655ancoms 259 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( E. s  e. 
Q.  ( s  e.  ( 2nd `  z
)  /\  s  e.  ( 1st `  y ) )  ->  z  <P  y ) )
57563adant1 933 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  ->  ( E. s  e.  Q.  ( s  e.  ( 2nd `  z )  /\  s  e.  ( 1st `  y ) )  ->  z  <P  y ) )
5857ad2antrr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P.  /\  z  e.  P. )  /\  s  e.  ( 1st `  y ) )  /\  s  e.  ( 2nd `  z
) )  ->  ( E. s  e.  Q.  ( s  e.  ( 2nd `  z )  /\  s  e.  ( 1st `  y ) )  ->  z  <P  y ) )
5953, 58mpd 13 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P.  /\  z  e.  P. )  /\  s  e.  ( 1st `  y ) )  /\  s  e.  ( 2nd `  z
) )  ->  z  <P  y )
6059ex 112 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  /\  s  e.  ( 1st `  y ) )  ->  ( s  e.  ( 2nd `  z
)  ->  z  <P  y ) )
6160adantrl 455 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  /\  ( r  e.  ( 2nd `  x )  /\  s  e.  ( 1st `  y ) ) )  ->  (
s  e.  ( 2nd `  z )  ->  z  <P  y ) )
6243, 61orim12d 710 . . . . . . . . . . . 12  |-  ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  /\  ( r  e.  ( 2nd `  x )  /\  s  e.  ( 1st `  y ) ) )  ->  (
( r  e.  ( 1st `  z )  \/  s  e.  ( 2nd `  z ) )  ->  ( x  <P  z  \/  z  <P 
y ) ) )
6362adantlr 454 . . . . . . . . . . 11  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P.  /\  z  e.  P. )  /\  ( q  e.  ( 2nd `  x
)  /\  q  e.  ( 1st `  y ) ) )  /\  (
r  e.  ( 2nd `  x )  /\  s  e.  ( 1st `  y
) ) )  -> 
( ( r  e.  ( 1st `  z
)  \/  s  e.  ( 2nd `  z
) )  ->  (
x  <P  z  \/  z  <P  y ) ) )
6463adantr 265 . . . . . . . . . 10  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e. 
P. )  /\  (
q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y
) ) )  /\  ( r  e.  ( 2nd `  x )  /\  s  e.  ( 1st `  y ) ) )  /\  (
r  <Q  q  /\  q  <Q  s ) )  -> 
( ( r  e.  ( 1st `  z
)  \/  s  e.  ( 2nd `  z
) )  ->  (
x  <P  z  \/  z  <P  y ) ) )
6526, 64mpd 13 . . . . . . . . 9  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e. 
P. )  /\  (
q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y
) ) )  /\  ( r  e.  ( 2nd `  x )  /\  s  e.  ( 1st `  y ) ) )  /\  (
r  <Q  q  /\  q  <Q  s ) )  -> 
( x  <P  z  \/  z  <P  y ) )
6665ex 112 . . . . . . . 8  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P.  /\  z  e.  P. )  /\  ( q  e.  ( 2nd `  x
)  /\  q  e.  ( 1st `  y ) ) )  /\  (
r  e.  ( 2nd `  x )  /\  s  e.  ( 1st `  y
) ) )  -> 
( ( r  <Q 
q  /\  q  <Q  s )  ->  ( x  <P  z  \/  z  <P 
y ) ) )
6766rexlimdvva 2457 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  /\  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) )  ->  ( E. r  e.  ( 2nd `  x ) E. s  e.  ( 1st `  y ) ( r 
<Q  q  /\  q  <Q  s )  ->  (
x  <P  z  \/  z  <P  y ) ) )
6814, 67mpd 13 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  /\  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) )  ->  (
x  <P  z  \/  z  <P  y ) )
6968ex 112 . . . . 5  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  ->  (
( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) )  ->  ( x  <P  z  \/  z  <P 
y ) ) )
7069rexlimdvw 2453 . . . 4  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  ->  ( E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) )  ->  ( x  <P  z  \/  z  <P 
y ) ) )
713, 70sylbid 143 . . 3  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  ->  (
x  <P  y  ->  (
x  <P  z  \/  z  <P  y ) ) )
7271rgen3 2423 . 2  |-  A. x  e.  P.  A. y  e. 
P.  A. z  e.  P.  ( x  <P  y  -> 
( x  <P  z  \/  z  <P  y ) )
73 df-iso 4062 . 2  |-  (  <P  Or  P.  <->  (  <P  Po  P.  /\ 
A. x  e.  P.  A. y  e.  P.  A. z  e.  P.  (
x  <P  y  ->  (
x  <P  z  \/  z  <P  y ) ) ) )
741, 72, 73mpbir2an 860 1  |-  <P  Or  P.
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639    /\ w3a 896   E.wex 1397    e. wcel 1409   A.wral 2323   E.wrex 2324   <.cop 3406   class class class wbr 3792    Po wpo 4059    Or wor 4060   ` cfv 4930   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436    <Q cltq 6441   P.cnp 6447    <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-mi 6462  df-lti 6463  df-enq 6503  df-nqqs 6504  df-ltnqqs 6509  df-inp 6622  df-iltp 6626
This theorem is referenced by:  prplnqu  6776  addextpr  6777  caucvgprprlemk  6839  caucvgprprlemnkltj  6845  caucvgprprlemnkeqj  6846  caucvgprprlemnjltk  6847  caucvgprprlemnbj  6849  caucvgprprlemml  6850  caucvgprprlemlol  6854  caucvgprprlemupu  6856  caucvgprprlemloc  6859  caucvgprprlemaddq  6864  lttrsr  6905  ltposr  6906  ltsosr  6907  archsr  6924
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