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Theorem ltsopr 7216
Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4133). 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 7215 . 2  |-  <P  Po  P.
2 ltdfpr 7126 . . . . 5  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  <P  y  <->  E. q  e.  Q.  (
q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y
) ) ) )
323adant3 964 . . . 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 7095 . . . . . . . . . . . 12  |-  ( x  e.  P.  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  P. )
5 prnminu 7109 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  P.  /\  q  e.  ( 2nd `  x ) )  ->  E. r  e.  ( 2nd `  x ) r 
<Q  q )
64, 5sylan 278 . . . . . . . . . . 11  |-  ( ( x  e.  P.  /\  q  e.  ( 2nd `  x ) )  ->  E. r  e.  ( 2nd `  x ) r 
<Q  q )
7 prop 7095 . . . . . . . . . . . 12  |-  ( y  e.  P.  ->  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  e.  P. )
8 prnmaxl 7108 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  P.  /\  q  e.  ( 1st `  y ) )  ->  E. s  e.  ( 1st `  y ) q 
<Q  s )
97, 8sylan 278 . . . . . . . . . . 11  |-  ( ( y  e.  P.  /\  q  e.  ( 1st `  y ) )  ->  E. s  e.  ( 1st `  y ) q 
<Q  s )
106, 9anim12i 332 . . . . . . . . . 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 556 . . . . . . . . 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 2537 . . . . . . . . 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 133 . . . . . . . 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 1102 . . . . . . 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 7018 . . . . . . . . . . . . 13  |-  <Q  Or  Q.
16 ltrelnq 6985 . . . . . . . . . . . . 13  |-  <Q  C_  ( Q.  X.  Q. )
1715, 16sotri 4840 . . . . . . . . . . . 12  |-  ( ( r  <Q  q  /\  q  <Q  s )  -> 
r  <Q  s )
1817adantl 272 . . . . . . . . . . 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 7095 . . . . . . . . . . . . . . . 16  |-  ( z  e.  P.  ->  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  e.  P. )
20 prloc 7111 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  z
) ,  ( 2nd `  z ) >.  e.  P.  /\  r  <Q  s )  ->  ( r  e.  ( 1st `  z )  \/  s  e.  ( 2nd `  z ) ) )
2119, 20sylan 278 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  P.  /\  r  <Q  s )  -> 
( r  e.  ( 1st `  z )  \/  s  e.  ( 2nd `  z ) ) )
22213ad2antl3 1108 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  /\  r  <Q  s )  ->  ( r  e.  ( 1st `  z
)  \/  s  e.  ( 2nd `  z
) ) )
2322ex 114 . . . . . . . . . . . . 13  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  ->  (
r  <Q  s  ->  (
r  e.  ( 1st `  z )  \/  s  e.  ( 2nd `  z
) ) ) )
2423adantr 271 . . . . . . . . . . . 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 473 . . . . . . . . . . 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 7102 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  P.  /\  r  e.  ( 2nd `  x ) )  -> 
r  e.  Q. )
284, 27sylan 278 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  P.  /\  r  e.  ( 2nd `  x ) )  -> 
r  e.  Q. )
29 ax-ia3 107 . . . . . . . . . . . . . . . . . . . . 21  |-  ( r  e.  ( 2nd `  x
)  ->  ( r  e.  ( 1st `  z
)  ->  ( r  e.  ( 2nd `  x
)  /\  r  e.  ( 1st `  z ) ) ) )
3029adantl 272 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  P.  /\  r  e.  ( 2nd `  x ) )  -> 
( r  e.  ( 1st `  z )  ->  ( r  e.  ( 2nd `  x
)  /\  r  e.  ( 1st `  z ) ) ) )
31 19.8a 1528 . . . . . . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . . . . . . 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 1106 . . . . . . . . . . . . . . . . . 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 123 . . . . . . . . . . . . . . . . 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 2366 . . . . . . . . . . . . . . . . 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 133 . . . . . . . . . . . . . . . 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 7126 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  <P  z  <->  E. r  e.  Q.  (
r  e.  ( 2nd `  x )  /\  r  e.  ( 1st `  z
) ) ) )
3837biimprd 157 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( E. r  e. 
Q.  ( r  e.  ( 2nd `  x
)  /\  r  e.  ( 1st `  z ) )  ->  x  <P  z ) )
39383adant2 963 . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . 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 114 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  /\  r  e.  ( 2nd `  x ) )  ->  ( r  e.  ( 1st `  z
)  ->  x  <P  z ) )
4342adantrr 464 . . . . . . . . . . . . 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 7101 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  P.  /\  s  e.  ( 1st `  y ) )  -> 
s  e.  Q. )
457, 44sylan 278 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  P.  /\  s  e.  ( 1st `  y ) )  -> 
s  e.  Q. )
46 pm3.21 261 . . . . . . . . . . . . . . . . . . . . 21  |-  ( s  e.  ( 1st `  y
)  ->  ( s  e.  ( 2nd `  z
)  ->  ( s  e.  ( 2nd `  z
)  /\  s  e.  ( 1st `  y ) ) ) )
4746adantl 272 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  P.  /\  s  e.  ( 1st `  y ) )  -> 
( s  e.  ( 2nd `  z )  ->  ( s  e.  ( 2nd `  z
)  /\  s  e.  ( 1st `  y ) ) ) )
48 19.8a 1528 . . . . . . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . . . . . . 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 1107 . . . . . . . . . . . . . . . . . 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 123 . . . . . . . . . . . . . . . . 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 2366 . . . . . . . . . . . . . . . . 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 133 . . . . . . . . . . . . . . . 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 7126 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  e.  P.  /\  y  e.  P. )  ->  ( z  <P  y  <->  E. s  e.  Q.  (
s  e.  ( 2nd `  z )  /\  s  e.  ( 1st `  y
) ) ) )
5554biimprd 157 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  e.  P.  /\  y  e.  P. )  ->  ( E. s  e. 
Q.  ( s  e.  ( 2nd `  z
)  /\  s  e.  ( 1st `  y ) )  ->  z  <P  y ) )
5655ancoms 265 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( E. s  e. 
Q.  ( s  e.  ( 2nd `  z
)  /\  s  e.  ( 1st `  y ) )  ->  z  <P  y ) )
57563adant1 962 . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . 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 114 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  /\  s  e.  ( 1st `  y ) )  ->  ( s  e.  ( 2nd `  z
)  ->  z  <P  y ) )
6160adantrl 463 . . . . . . . . . . . . 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 736 . . . . . . . . . . . 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 462 . . . . . . . . . . 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 271 . . . . . . . . . 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 114 . . . . . . . 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 2497 . . . . . . 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 114 . . . . 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 2493 . . . 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 149 . . 3  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  z  e.  P. )  ->  (
x  <P  y  ->  (
x  <P  z  \/  z  <P  y ) ) )
7271rgen3 2461 . 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 4133 . 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 889 1  |-  <P  Or  P.
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 665    /\ w3a 925   E.wex 1427    e. wcel 1439   A.wral 2360   E.wrex 2361   <.cop 3453   class class class wbr 3851    Po wpo 4130    Or wor 4131   ` cfv 5028   1stc1st 5923   2ndc2nd 5924   Q.cnq 6900    <Q cltq 6905   P.cnp 6911    <P cltp 6915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-mi 6926  df-lti 6927  df-enq 6967  df-nqqs 6968  df-ltnqqs 6973  df-inp 7086  df-iltp 7090
This theorem is referenced by:  prplnqu  7240  addextpr  7241  caucvgprprlemk  7303  caucvgprprlemnkltj  7309  caucvgprprlemnkeqj  7310  caucvgprprlemnjltk  7311  caucvgprprlemnbj  7313  caucvgprprlemml  7314  caucvgprprlemlol  7318  caucvgprprlemupu  7320  caucvgprprlemloc  7323  caucvgprprlemaddq  7328  lttrsr  7369  ltposr  7370  ltsosr  7371  archsr  7388
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