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Theorem isibg2aalem2 26114
Description: Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)
Assertion
Ref Expression
isibg2aalem2  |-  ( Z  e.  P  ->  ( A. z  e.  P  ( ( ( { X ,  Y , 
z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z
) )  ->  (
( X  e.  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B z )  ->  ( Y  e.  ( z B X )  /\  { X ,  Y , 
z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z
) ) )  /\  A. l  e.  L  ( ( X  e/  l  /\  Y  e/  l  /\  z  e/  l
)  ->  ( (
( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B z )  i^i  l
)  =  (/) )  -> 
( ( X B z )  i^i  l
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B z )  i^i  l )  =/=  (/) )  -> 
( ( X B z )  i^i  l
)  =  (/) ) ) ) ) )  -> 
( ( ( { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z
) )  ->  (
( X  e.  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z )  /\  Y  e.  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B Z )  ->  ( Y  e.  ( Z B X )  /\  { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z
) ) )  /\  A. l  e.  L  ( ( X  e/  l  /\  Y  e/  l  /\  Z  e/  l
)  ->  ( (
( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l
)  =  (/) )  -> 
( ( X B Z )  i^i  l
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) )  -> 
( ( X B Z )  i^i  l
)  =  (/) ) ) ) ) ) ) )
Distinct variable groups:    z, l, Z    z, B    z, C    z, L    z, P    z, X    z, Y
Allowed substitution hints:    B( l)    C( l)    P( l)    L( l)    X( l)    Y( l)

Proof of Theorem isibg2aalem2
StepHypRef Expression
1 tpeq3 3717 . . . . . 6  |-  ( z  =  Z  ->  { X ,  Y ,  z }  =  { X ,  Y ,  Z }
)
21eleq1d 2349 . . . . 5  |-  ( z  =  Z  ->  ( { X ,  Y , 
z }  e.  C  <->  { X ,  Y ,  Z }  e.  C
) )
3 neeq2 2455 . . . . . 6  |-  ( z  =  Z  ->  ( X  =/=  z  <->  X  =/=  Z ) )
4 neeq2 2455 . . . . . 6  |-  ( z  =  Z  ->  ( Y  =/=  z  <->  Y  =/=  Z ) )
53, 43anbi23d 1255 . . . . 5  |-  ( z  =  Z  ->  (
( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z
)  <->  ( X  =/= 
Y  /\  X  =/=  Z  /\  Y  =/=  Z
) ) )
62, 5anbi12d 691 . . . 4  |-  ( z  =  Z  ->  (
( { X ,  Y ,  z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) )  <->  ( { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z
) ) ) )
7 oveq2 5866 . . . . . . 7  |-  ( z  =  Z  ->  ( Y B z )  =  ( Y B Z ) )
87eleq2d 2350 . . . . . 6  |-  ( z  =  Z  ->  ( X  e.  ( Y B z )  <->  X  e.  ( Y B Z ) ) )
9 eqidd 2284 . . . . . . 7  |-  ( z  =  Z  ->  Y  =  Y )
10 oveq2 5866 . . . . . . 7  |-  ( z  =  Z  ->  ( X B z )  =  ( X B Z ) )
119, 10neleq12d 24933 . . . . . 6  |-  ( z  =  Z  ->  ( Y  e/  ( X B z )  <->  Y  e/  ( X B Z ) ) )
12 neleq1 2537 . . . . . 6  |-  ( z  =  Z  ->  (
z  e/  ( X B Y )  <->  Z  e/  ( X B Y ) ) )
138, 11, 123anbi123d 1252 . . . . 5  |-  ( z  =  Z  ->  (
( X  e.  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  <->  ( X  e.  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e/  ( X B Y ) ) ) )
14 eqidd 2284 . . . . . . 7  |-  ( z  =  Z  ->  X  =  X )
1514, 7neleq12d 24933 . . . . . 6  |-  ( z  =  Z  ->  ( X  e/  ( Y B z )  <->  X  e/  ( Y B Z ) ) )
1610eleq2d 2350 . . . . . 6  |-  ( z  =  Z  ->  ( Y  e.  ( X B z )  <->  Y  e.  ( X B Z ) ) )
1715, 16, 123anbi123d 1252 . . . . 5  |-  ( z  =  Z  ->  (
( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  <->  ( X  e/  ( Y B Z )  /\  Y  e.  ( X B Z )  /\  Z  e/  ( X B Y ) ) ) )
18 eleq1 2343 . . . . . 6  |-  ( z  =  Z  ->  (
z  e.  ( X B Y )  <->  Z  e.  ( X B Y ) ) )
1915, 11, 183anbi123d 1252 . . . . 5  |-  ( z  =  Z  ->  (
( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) )  <->  ( X  e/  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e.  ( X B Y ) ) ) )
2013, 17, 193orbi123d 1251 . . . 4  |-  ( z  =  Z  ->  (
( ( X  e.  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) )  <-> 
( ( X  e.  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z )  /\  Y  e.  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e.  ( X B Y ) ) ) ) )
216, 20imbi12d 311 . . 3  |-  ( z  =  Z  ->  (
( ( { X ,  Y ,  z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) )  -> 
( ( X  e.  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) ) )  <->  ( ( { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z
) )  ->  (
( X  e.  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z )  /\  Y  e.  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e.  ( X B Y ) ) ) ) ) )
22 oveq1 5865 . . . . . . 7  |-  ( z  =  Z  ->  (
z B X )  =  ( Z B X ) )
2322eleq2d 2350 . . . . . 6  |-  ( z  =  Z  ->  ( Y  e.  ( z B X )  <->  Y  e.  ( Z B X ) ) )
2423, 2, 53anbi123d 1252 . . . . 5  |-  ( z  =  Z  ->  (
( Y  e.  ( z B X )  /\  { X ,  Y ,  z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) )  <->  ( Y  e.  ( Z B X )  /\  { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) ) ) )
2516, 24imbi12d 311 . . . 4  |-  ( z  =  Z  ->  (
( Y  e.  ( X B z )  ->  ( Y  e.  ( z B X )  /\  { X ,  Y ,  z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) ) )  <-> 
( Y  e.  ( X B Z )  ->  ( Y  e.  ( Z B X )  /\  { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) ) ) ) )
26 neleq1 2537 . . . . . . 7  |-  ( z  =  Z  ->  (
z  e/  l  <->  Z  e/  l ) )
27263anbi3d 1258 . . . . . 6  |-  ( z  =  Z  ->  (
( X  e/  l  /\  Y  e/  l  /\  z  e/  l
)  <->  ( X  e/  l  /\  Y  e/  l  /\  Z  e/  l
) ) )
287ineq1d 3369 . . . . . . . . . 10  |-  ( z  =  Z  ->  (
( Y B z )  i^i  l )  =  ( ( Y B Z )  i^i  l ) )
2928eqeq1d 2291 . . . . . . . . 9  |-  ( z  =  Z  ->  (
( ( Y B z )  i^i  l
)  =  (/)  <->  ( ( Y B Z )  i^i  l )  =  (/) ) )
3029anbi2d 684 . . . . . . . 8  |-  ( z  =  Z  ->  (
( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B z )  i^i  l
)  =  (/) )  <->  ( (
( X B Y )  i^i  l )  =  (/)  /\  (
( Y B Z )  i^i  l )  =  (/) ) ) )
3110ineq1d 3369 . . . . . . . . 9  |-  ( z  =  Z  ->  (
( X B z )  i^i  l )  =  ( ( X B Z )  i^i  l ) )
3231eqeq1d 2291 . . . . . . . 8  |-  ( z  =  Z  ->  (
( ( X B z )  i^i  l
)  =  (/)  <->  ( ( X B Z )  i^i  l )  =  (/) ) )
3330, 32imbi12d 311 . . . . . . 7  |-  ( z  =  Z  ->  (
( ( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B z )  i^i  l )  =  (/) )  ->  ( ( X B z )  i^i  l )  =  (/) ) 
<->  ( ( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l )  =  (/) )  ->  ( ( X B Z )  i^i  l )  =  (/) ) ) )
3428neeq1d 2459 . . . . . . . . 9  |-  ( z  =  Z  ->  (
( ( Y B z )  i^i  l
)  =/=  (/)  <->  ( ( Y B Z )  i^i  l )  =/=  (/) ) )
3534anbi2d 684 . . . . . . . 8  |-  ( z  =  Z  ->  (
( ( ( X B Y )  i^i  l )  =/=  (/)  /\  (
( Y B z )  i^i  l )  =/=  (/) )  <->  ( (
( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) ) ) )
3635, 32imbi12d 311 . . . . . . 7  |-  ( z  =  Z  ->  (
( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B z )  i^i  l )  =/=  (/) )  -> 
( ( X B z )  i^i  l
)  =  (/) )  <->  ( (
( ( X B Y )  i^i  l
)  =/=  (/)  /\  (
( Y B Z )  i^i  l )  =/=  (/) )  ->  (
( X B Z )  i^i  l )  =  (/) ) ) )
3733, 36anbi12d 691 . . . . . 6  |-  ( z  =  Z  ->  (
( ( ( ( ( X B Y )  i^i  l )  =  (/)  /\  (
( Y B z )  i^i  l )  =  (/) )  ->  (
( X B z )  i^i  l )  =  (/) )  /\  (
( ( ( X B Y )  i^i  l )  =/=  (/)  /\  (
( Y B z )  i^i  l )  =/=  (/) )  ->  (
( X B z )  i^i  l )  =  (/) ) )  <->  ( (
( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l
)  =  (/) )  -> 
( ( X B Z )  i^i  l
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) )  -> 
( ( X B Z )  i^i  l
)  =  (/) ) ) ) )
3827, 37imbi12d 311 . . . . 5  |-  ( z  =  Z  ->  (
( ( X  e/  l  /\  Y  e/  l  /\  z  e/  l
)  ->  ( (
( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B z )  i^i  l
)  =  (/) )  -> 
( ( X B z )  i^i  l
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B z )  i^i  l )  =/=  (/) )  -> 
( ( X B z )  i^i  l
)  =  (/) ) ) )  <->  ( ( X  e/  l  /\  Y  e/  l  /\  Z  e/  l )  ->  (
( ( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l )  =  (/) )  ->  ( ( X B Z )  i^i  l )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) )  ->  ( ( X B Z )  i^i  l )  =  (/) ) ) ) ) )
3938ralbidv 2563 . . . 4  |-  ( z  =  Z  ->  ( A. l  e.  L  ( ( X  e/  l  /\  Y  e/  l  /\  z  e/  l
)  ->  ( (
( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B z )  i^i  l
)  =  (/) )  -> 
( ( X B z )  i^i  l
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B z )  i^i  l )  =/=  (/) )  -> 
( ( X B z )  i^i  l
)  =  (/) ) ) )  <->  A. l  e.  L  ( ( X  e/  l  /\  Y  e/  l  /\  Z  e/  l
)  ->  ( (
( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l
)  =  (/) )  -> 
( ( X B Z )  i^i  l
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) )  -> 
( ( X B Z )  i^i  l
)  =  (/) ) ) ) ) )
4025, 39anbi12d 691 . . 3  |-  ( z  =  Z  ->  (
( ( Y  e.  ( X B z )  ->  ( Y  e.  ( z B X )  /\  { X ,  Y ,  z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) ) )  /\  A. l  e.  L  ( ( X  e/  l  /\  Y  e/  l  /\  z  e/  l )  ->  (
( ( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B z )  i^i  l )  =  (/) )  ->  ( ( X B z )  i^i  l )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B z )  i^i  l )  =/=  (/) )  ->  ( ( X B z )  i^i  l )  =  (/) ) ) ) )  <-> 
( ( Y  e.  ( X B Z )  ->  ( Y  e.  ( Z B X )  /\  { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) ) )  /\  A. l  e.  L  ( ( X  e/  l  /\  Y  e/  l  /\  Z  e/  l )  ->  (
( ( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l )  =  (/) )  ->  ( ( X B Z )  i^i  l )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) )  ->  ( ( X B Z )  i^i  l )  =  (/) ) ) ) ) ) )
4121, 40anbi12d 691 . 2  |-  ( z  =  Z  ->  (
( ( ( { X ,  Y , 
z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z
) )  ->  (
( X  e.  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B z )  ->  ( Y  e.  ( z B X )  /\  { X ,  Y , 
z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z
) ) )  /\  A. l  e.  L  ( ( X  e/  l  /\  Y  e/  l  /\  z  e/  l
)  ->  ( (
( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B z )  i^i  l
)  =  (/) )  -> 
( ( X B z )  i^i  l
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B z )  i^i  l )  =/=  (/) )  -> 
( ( X B z )  i^i  l
)  =  (/) ) ) ) ) )  <->  ( (
( { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) )  -> 
( ( X  e.  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z )  /\  Y  e.  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B Z )  -> 
( Y  e.  ( Z B X )  /\  { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/= 
Z ) ) )  /\  A. l  e.  L  ( ( X  e/  l  /\  Y  e/  l  /\  Z  e/  l )  ->  (
( ( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l )  =  (/) )  ->  ( ( X B Z )  i^i  l )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) )  ->  ( ( X B Z )  i^i  l )  =  (/) ) ) ) ) ) ) )
4241rspcv 2880 1  |-  ( Z  e.  P  ->  ( A. z  e.  P  ( ( ( { X ,  Y , 
z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z
) )  ->  (
( X  e.  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B z )  ->  ( Y  e.  ( z B X )  /\  { X ,  Y , 
z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z
) ) )  /\  A. l  e.  L  ( ( X  e/  l  /\  Y  e/  l  /\  z  e/  l
)  ->  ( (
( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B z )  i^i  l
)  =  (/) )  -> 
( ( X B z )  i^i  l
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B z )  i^i  l )  =/=  (/) )  -> 
( ( X B z )  i^i  l
)  =  (/) ) ) ) ) )  -> 
( ( ( { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z
) )  ->  (
( X  e.  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z )  /\  Y  e.  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B Z )  ->  ( Y  e.  ( Z B X )  /\  { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z
) ) )  /\  A. l  e.  L  ( ( X  e/  l  /\  Y  e/  l  /\  Z  e/  l
)  ->  ( (
( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l
)  =  (/) )  -> 
( ( X B Z )  i^i  l
)  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) )  -> 
( ( X B Z )  i^i  l
)  =  (/) ) ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    e/ wnel 2447   A.wral 2543    i^i cin 3151   (/)c0 3455   {ctp 3642  (class class class)co 5858
This theorem is referenced by:  isib2g1a1  26116  isibg1a2  26117  isibg1a3a  26122  isibg1spa  26123  isibg1a5a  26124  bsstr  26128  nbssntr  26129
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 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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