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Theorem 3oalem2 23165
Description: Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
3oalem1.1  |-  B  e. 
CH
3oalem1.2  |-  C  e. 
CH
3oalem1.3  |-  R  e. 
CH
3oalem1.4  |-  S  e. 
CH
Assertion
Ref Expression
3oalem2  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  v  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) )
Distinct variable groups:    x, y,
z, w, v, B   
x, C, y, z, w, v    x, R, y, z, w, v   
x, S, y, z, w, v

Proof of Theorem 3oalem2
StepHypRef Expression
1 simplll 735 . . 3  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  x  e.  B
)
2 simpllr 736 . . . 4  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  y  e.  R
)
3 3oalem1.1 . . . . . . 7  |-  B  e. 
CH
4 3oalem1.2 . . . . . . 7  |-  C  e. 
CH
5 3oalem1.3 . . . . . . 7  |-  R  e. 
CH
6 3oalem1.4 . . . . . . 7  |-  S  e. 
CH
73, 4, 5, 63oalem1 23164 . . . . . 6  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  v  e.  ~H )  /\  ( z  e. 
~H  /\  w  e.  ~H ) ) )
8 hvaddsub12 22540 . . . . . . . . . 10  |-  ( ( y  e.  ~H  /\  w  e.  ~H  /\  w  e.  ~H )  ->  (
y  +h  ( w  -h  w ) )  =  ( w  +h  ( y  -h  w
) ) )
983anidm23 1243 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( y  +h  (
w  -h  w ) )  =  ( w  +h  ( y  -h  w ) ) )
10 hvsubid 22528 . . . . . . . . . . 11  |-  ( w  e.  ~H  ->  (
w  -h  w )  =  0h )
1110oveq2d 6097 . . . . . . . . . 10  |-  ( w  e.  ~H  ->  (
y  +h  ( w  -h  w ) )  =  ( y  +h 
0h ) )
12 ax-hvaddid 22507 . . . . . . . . . 10  |-  ( y  e.  ~H  ->  (
y  +h  0h )  =  y )
1311, 12sylan9eqr 2490 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( y  +h  (
w  -h  w ) )  =  y )
149, 13eqtr3d 2470 . . . . . . . 8  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( w  +h  (
y  -h  w ) )  =  y )
1514ad2ant2l 727 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( w  +h  ( y  -h  w
) )  =  y )
1615adantlr 696 . . . . . 6  |-  ( ( ( ( x  e. 
~H  /\  y  e.  ~H )  /\  v  e.  ~H )  /\  (
z  e.  ~H  /\  w  e.  ~H )
)  ->  ( w  +h  ( y  -h  w
) )  =  y )
177, 16syl 16 . . . . 5  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( w  +h  ( y  -h  w
) )  =  y )
18 simprlr 740 . . . . . 6  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  w  e.  S
)
19 eqtr2 2454 . . . . . . . . . . 11  |-  ( ( v  =  ( x  +h  y )  /\  v  =  ( z  +h  w ) )  -> 
( x  +h  y
)  =  ( z  +h  w ) )
2019oveq1d 6096 . . . . . . . . . 10  |-  ( ( v  =  ( x  +h  y )  /\  v  =  ( z  +h  w ) )  -> 
( ( x  +h  y )  -h  (
x  +h  w ) )  =  ( ( z  +h  w )  -h  ( x  +h  w ) ) )
2120ad2ant2l 727 . . . . . . . . 9  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( ( x  +h  y )  -h  ( x  +h  w
) )  =  ( ( z  +h  w
)  -h  ( x  +h  w ) ) )
22 simpl 444 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  x  e.  ~H )
2322anim1i 552 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( x  e. 
~H  /\  w  e.  ~H ) )
24 hvsub4 22539 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( x  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
x  +h  y )  -h  ( x  +h  w ) )  =  ( ( x  -h  x )  +h  (
y  -h  w ) ) )
2523, 24syldan 457 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x  +h  y )  -h  ( x  +h  w
) )  =  ( ( x  -h  x
)  +h  ( y  -h  w ) ) )
26 hvsubid 22528 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  (
x  -h  x )  =  0h )
2726ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( x  -h  x )  =  0h )
2827oveq1d 6096 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x  -h  x )  +h  ( y  -h  w
) )  =  ( 0h  +h  ( y  -h  w ) ) )
29 hvsubcl 22520 . . . . . . . . . . . . . 14  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( y  -h  w
)  e.  ~H )
30 hvaddid2 22525 . . . . . . . . . . . . . 14  |-  ( ( y  -h  w )  e.  ~H  ->  ( 0h  +h  ( y  -h  w ) )  =  ( y  -h  w
) )
3129, 30syl 16 . . . . . . . . . . . . 13  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( 0h  +h  (
y  -h  w ) )  =  ( y  -h  w ) )
3231adantll 695 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( 0h  +h  ( y  -h  w
) )  =  ( y  -h  w ) )
3325, 28, 323eqtrd 2472 . . . . . . . . . . 11  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x  +h  y )  -h  ( x  +h  w
) )  =  ( y  -h  w ) )
3433ad2ant2rl 730 . . . . . . . . . 10  |-  ( ( ( ( x  e. 
~H  /\  y  e.  ~H )  /\  v  e.  ~H )  /\  (
z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
x  +h  y )  -h  ( x  +h  w ) )  =  ( y  -h  w
) )
357, 34syl 16 . . . . . . . . 9  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( ( x  +h  y )  -h  ( x  +h  w
) )  =  ( y  -h  w ) )
36 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( z  e.  ~H  /\  w  e. 
~H ) )
37 simpr 448 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ~H  /\  w  e.  ~H )  ->  w  e.  ~H )
3837anim2i 553 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( x  e.  ~H  /\  w  e. 
~H ) )
39 hvsub4 22539 . . . . . . . . . . . . . 14  |-  ( ( ( z  e.  ~H  /\  w  e.  ~H )  /\  ( x  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( ( z  -h  x )  +h  (
w  -h  w ) ) )
4036, 38, 39syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( ( z  -h  x )  +h  (
w  -h  w ) ) )
4110ad2antll 710 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( w  -h  w )  =  0h )
4241oveq2d 6097 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  -h  x )  +h  ( w  -h  w ) )  =  ( ( z  -h  x )  +h  0h ) )
43 hvsubcl 22520 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ~H  /\  x  e.  ~H )  ->  ( z  -h  x
)  e.  ~H )
44 ax-hvaddid 22507 . . . . . . . . . . . . . . . 16  |-  ( ( z  -h  x )  e.  ~H  ->  (
( z  -h  x
)  +h  0h )  =  ( z  -h  x ) )
4543, 44syl 16 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ~H  /\  x  e.  ~H )  ->  ( ( z  -h  x )  +h  0h )  =  ( z  -h  x ) )
4645ancoms 440 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( z  -h  x )  +h  0h )  =  ( z  -h  x ) )
4746adantrr 698 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  -h  x )  +h  0h )  =  ( z  -h  x
) )
4840, 42, 473eqtrd 2472 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( z  -h  x
) )
4948adantlr 696 . . . . . . . . . . 11  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( z  -h  x
) )
5049adantlr 696 . . . . . . . . . 10  |-  ( ( ( ( x  e. 
~H  /\  y  e.  ~H )  /\  v  e.  ~H )  /\  (
z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( z  -h  x
) )
517, 50syl 16 . . . . . . . . 9  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( ( z  +h  w )  -h  ( x  +h  w
) )  =  ( z  -h  x ) )
5221, 35, 513eqtr3d 2476 . . . . . . . 8  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( y  -h  w )  =  ( z  -h  x ) )
53 simpll 731 . . . . . . . . 9  |-  ( ( ( x  e.  B  /\  y  e.  R
)  /\  v  =  ( x  +h  y
) )  ->  x  e.  B )
54 simpll 731 . . . . . . . . 9  |-  ( ( ( z  e.  C  /\  w  e.  S
)  /\  v  =  ( z  +h  w
) )  ->  z  e.  C )
554chshii 22730 . . . . . . . . . . . 12  |-  C  e.  SH
563chshii 22730 . . . . . . . . . . . 12  |-  B  e.  SH
5755, 56shsvsi 22869 . . . . . . . . . . 11  |-  ( ( z  e.  C  /\  x  e.  B )  ->  ( z  -h  x
)  e.  ( C  +H  B ) )
5857ancoms 440 . . . . . . . . . 10  |-  ( ( x  e.  B  /\  z  e.  C )  ->  ( z  -h  x
)  e.  ( C  +H  B ) )
5956, 55shscomi 22865 . . . . . . . . . 10  |-  ( B  +H  C )  =  ( C  +H  B
)
6058, 59syl6eleqr 2527 . . . . . . . . 9  |-  ( ( x  e.  B  /\  z  e.  C )  ->  ( z  -h  x
)  e.  ( B  +H  C ) )
6153, 54, 60syl2an 464 . . . . . . . 8  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( z  -h  x )  e.  ( B  +H  C ) )
6252, 61eqeltrd 2510 . . . . . . 7  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( y  -h  w )  e.  ( B  +H  C ) )
63 simplr 732 . . . . . . . 8  |-  ( ( ( x  e.  B  /\  y  e.  R
)  /\  v  =  ( x  +h  y
) )  ->  y  e.  R )
64 simplr 732 . . . . . . . 8  |-  ( ( ( z  e.  C  /\  w  e.  S
)  /\  v  =  ( z  +h  w
) )  ->  w  e.  S )
655chshii 22730 . . . . . . . . 9  |-  R  e.  SH
666chshii 22730 . . . . . . . . 9  |-  S  e.  SH
6765, 66shsvsi 22869 . . . . . . . 8  |-  ( ( y  e.  R  /\  w  e.  S )  ->  ( y  -h  w
)  e.  ( R  +H  S ) )
6863, 64, 67syl2an 464 . . . . . . 7  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( y  -h  w )  e.  ( R  +H  S ) )
69 elin 3530 . . . . . . 7  |-  ( ( y  -h  w )  e.  ( ( B  +H  C )  i^i  ( R  +H  S
) )  <->  ( (
y  -h  w )  e.  ( B  +H  C )  /\  (
y  -h  w )  e.  ( R  +H  S ) ) )
7062, 68, 69sylanbrc 646 . . . . . 6  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( y  -h  w )  e.  ( ( B  +H  C
)  i^i  ( R  +H  S ) ) )
7156, 55shscli 22819 . . . . . . . 8  |-  ( B  +H  C )  e.  SH
7265, 66shscli 22819 . . . . . . . 8  |-  ( R  +H  S )  e.  SH
7371, 72shincli 22864 . . . . . . 7  |-  ( ( B  +H  C )  i^i  ( R  +H  S ) )  e.  SH
7466, 73shsvai 22866 . . . . . 6  |-  ( ( w  e.  S  /\  ( y  -h  w
)  e.  ( ( B  +H  C )  i^i  ( R  +H  S ) ) )  ->  ( w  +h  ( y  -h  w
) )  e.  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) )
7518, 70, 74syl2anc 643 . . . . 5  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( w  +h  ( y  -h  w
) )  e.  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) )
7617, 75eqeltrrd 2511 . . . 4  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  y  e.  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) )
77 elin 3530 . . . 4  |-  ( y  e.  ( R  i^i  ( S  +H  (
( B  +H  C
)  i^i  ( R  +H  S ) ) ) )  <->  ( y  e.  R  /\  y  e.  ( S  +H  (
( B  +H  C
)  i^i  ( R  +H  S ) ) ) ) )
782, 76, 77sylanbrc 646 . . 3  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  y  e.  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S
) ) ) ) )
7966, 73shscli 22819 . . . . 5  |-  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S
) ) )  e.  SH
8065, 79shincli 22864 . . . 4  |-  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) )  e.  SH
8156, 80shsvai 22866 . . 3  |-  ( ( x  e.  B  /\  y  e.  ( R  i^i  ( S  +H  (
( B  +H  C
)  i^i  ( R  +H  S ) ) ) ) )  ->  (
x  +h  y )  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S
) ) ) ) ) )
821, 78, 81syl2anc 643 . 2  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( x  +h  y )  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) )
83 eleq1 2496 . . 3  |-  ( v  =  ( x  +h  y )  ->  (
v  e.  ( B  +H  ( R  i^i  ( S  +H  (
( B  +H  C
)  i^i  ( R  +H  S ) ) ) ) )  <->  ( x  +h  y )  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) ) )
8483ad2antlr 708 . 2  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( v  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) )  <->  ( x  +h  y )  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) ) )
8582, 84mpbird 224 1  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  v  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3319  (class class class)co 6081   ~Hchil 22422    +h cva 22423   0hc0v 22427    -h cmv 22428   CHcch 22432    +H cph 22434
This theorem is referenced by:  3oalem3  23166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-hilex 22502  ax-hfvadd 22503  ax-hvcom 22504  ax-hvass 22505  ax-hv0cl 22506  ax-hvaddid 22507  ax-hfvmul 22508  ax-hvmulid 22509  ax-hvdistr1 22511  ax-hvdistr2 22512  ax-hvmul0 22513
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-neg 9294  df-nn 10001  df-grpo 21779  df-ablo 21870  df-hvsub 22474  df-hlim 22475  df-sh 22709  df-ch 22724  df-shs 22810
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