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Theorem cdaassen 7692
Description: Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
cdaassen  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  +c  B )  +c  C
)  ~~  ( A  +c  ( B  +c  C
) ) )

Proof of Theorem cdaassen
StepHypRef Expression
1 simp1 960 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  e.  V )
2 0ex 4047 . . . . . 6  |-  (/)  e.  _V
3 xpsneng 6832 . . . . . 6  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
41, 2, 3sylancl 646 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  { (/)
} )  ~~  A
)
5 ensym 6796 . . . . 5  |-  ( ( A  X.  { (/) } )  ~~  A  ->  A  ~~  ( A  X.  { (/) } ) )
64, 5syl 17 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  ~~  ( A  X.  { (/) } ) )
7 simp2 961 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  e.  W )
8 snex 4110 . . . . . . . 8  |-  { (/) }  e.  _V
9 xpexg 4707 . . . . . . . 8  |-  ( ( B  e.  W  /\  {
(/) }  e.  _V )  ->  ( B  X.  { (/) } )  e. 
_V )
107, 8, 9sylancl 646 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  e.  _V )
11 1on 6372 . . . . . . 7  |-  1o  e.  On
12 xpsneng 6832 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  e.  _V  /\  1o  e.  On )  ->  ( ( B  X.  { (/) } )  X.  { 1o }
)  ~~  ( B  X.  { (/) } ) )
1310, 11, 12sylancl 646 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( B  X.  { (/) } )  X. 
{ 1o } ) 
~~  ( B  X.  { (/) } ) )
14 xpsneng 6832 . . . . . . 7  |-  ( ( B  e.  W  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
157, 2, 14sylancl 646 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  ~~  B
)
16 entr 6798 . . . . . 6  |-  ( ( ( ( B  X.  { (/) } )  X. 
{ 1o } ) 
~~  ( B  X.  { (/) } )  /\  ( B  X.  { (/) } )  ~~  B )  ->  ( ( B  X.  { (/) } )  X.  { 1o }
)  ~~  B )
1713, 15, 16syl2anc 645 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( B  X.  { (/) } )  X. 
{ 1o } ) 
~~  B )
18 ensym 6796 . . . . 5  |-  ( ( ( B  X.  { (/)
} )  X.  { 1o } )  ~~  B  ->  B  ~~  ( ( B  X.  { (/) } )  X.  { 1o } ) )
1917, 18syl 17 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  ~~  ( ( B  X.  { (/) } )  X.  { 1o } ) )
20 xp01disj 6381 . . . . 5  |-  ( ( A  X.  { (/) } )  i^i  ( ( B  X.  { (/) } )  X.  { 1o } ) )  =  (/)
2120a1i 12 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  X.  { (/) } )  i^i  ( ( B  X.  { (/) } )  X. 
{ 1o } ) )  =  (/) )
22 cdaenun 7684 . . . 4  |-  ( ( A  ~~  ( A  X.  { (/) } )  /\  B  ~~  (
( B  X.  { (/)
} )  X.  { 1o } )  /\  (
( A  X.  { (/)
} )  i^i  (
( B  X.  { (/)
} )  X.  { 1o } ) )  =  (/) )  ->  ( A  +c  B )  ~~  ( ( A  X.  { (/) } )  u.  ( ( B  X.  { (/) } )  X. 
{ 1o } ) ) )
236, 19, 21, 22syl3anc 1187 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  +c  B
)  ~~  ( ( A  X.  { (/) } )  u.  ( ( B  X.  { (/) } )  X.  { 1o }
) ) )
24 simp3 962 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  e.  X )
25 snex 4110 . . . . . . 7  |-  { 1o }  e.  _V
26 xpexg 4707 . . . . . . 7  |-  ( ( C  e.  X  /\  { 1o }  e.  _V )  ->  ( C  X.  { 1o } )  e. 
_V )
2724, 25, 26sylancl 646 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  e.  _V )
28 xpsneng 6832 . . . . . 6  |-  ( ( ( C  X.  { 1o } )  e.  _V  /\  1o  e.  On )  ->  ( ( C  X.  { 1o }
)  X.  { 1o } )  ~~  ( C  X.  { 1o }
) )
2927, 11, 28sylancl 646 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( C  X.  { 1o } )  X. 
{ 1o } ) 
~~  ( C  X.  { 1o } ) )
30 xpsneng 6832 . . . . . 6  |-  ( ( C  e.  X  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
3124, 11, 30sylancl 646 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  ~~  C
)
32 entr 6798 . . . . 5  |-  ( ( ( ( C  X.  { 1o } )  X. 
{ 1o } ) 
~~  ( C  X.  { 1o } )  /\  ( C  X.  { 1o } )  ~~  C
)  ->  ( ( C  X.  { 1o }
)  X.  { 1o } )  ~~  C
)
3329, 31, 32syl2anc 645 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( C  X.  { 1o } )  X. 
{ 1o } ) 
~~  C )
34 ensym 6796 . . . 4  |-  ( ( ( C  X.  { 1o } )  X.  { 1o } )  ~~  C  ->  C  ~~  ( ( C  X.  { 1o } )  X.  { 1o } ) )
3533, 34syl 17 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  ~~  ( ( C  X.  { 1o } )  X.  { 1o } ) )
36 indir 3324 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  (
( B  X.  { (/)
} )  X.  { 1o } ) )  i^i  ( ( C  X.  { 1o } )  X. 
{ 1o } ) )  =  ( ( ( A  X.  { (/)
} )  i^i  (
( C  X.  { 1o } )  X.  { 1o } ) )  u.  ( ( ( B  X.  { (/) } )  X.  { 1o }
)  i^i  ( ( C  X.  { 1o }
)  X.  { 1o } ) ) )
37 xp01disj 6381 . . . . . 6  |-  ( ( A  X.  { (/) } )  i^i  ( ( C  X.  { 1o } )  X.  { 1o } ) )  =  (/)
38 xp01disj 6381 . . . . . . . 8  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
3938xpeq1i 4616 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  i^i  ( C  X.  { 1o }
) )  X.  { 1o } )  =  (
(/)  X.  { 1o } )
40 xpindir 4727 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  i^i  ( C  X.  { 1o }
) )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  X. 
{ 1o } )  i^i  ( ( C  X.  { 1o }
)  X.  { 1o } ) )
41 xp0r 4675 . . . . . . 7  |-  ( (/)  X. 
{ 1o } )  =  (/)
4239, 40, 413eqtr3i 2281 . . . . . 6  |-  ( ( ( B  X.  { (/)
} )  X.  { 1o } )  i^i  (
( C  X.  { 1o } )  X.  { 1o } ) )  =  (/)
4337, 42uneq12i 3237 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  i^i  (
( C  X.  { 1o } )  X.  { 1o } ) )  u.  ( ( ( B  X.  { (/) } )  X.  { 1o }
)  i^i  ( ( C  X.  { 1o }
)  X.  { 1o } ) ) )  =  ( (/)  u.  (/) )
44 un0 3386 . . . . 5  |-  ( (/)  u.  (/) )  =  (/)
4536, 43, 443eqtri 2277 . . . 4  |-  ( ( ( A  X.  { (/)
} )  u.  (
( B  X.  { (/)
} )  X.  { 1o } ) )  i^i  ( ( C  X.  { 1o } )  X. 
{ 1o } ) )  =  (/)
4645a1i 12 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( ( A  X.  { (/) } )  u.  ( ( B  X.  { (/) } )  X.  { 1o }
) )  i^i  (
( C  X.  { 1o } )  X.  { 1o } ) )  =  (/) )
47 cdaenun 7684 . . 3  |-  ( ( ( A  +c  B
)  ~~  ( ( A  X.  { (/) } )  u.  ( ( B  X.  { (/) } )  X.  { 1o }
) )  /\  C  ~~  ( ( C  X.  { 1o } )  X. 
{ 1o } )  /\  ( ( ( A  X.  { (/) } )  u.  ( ( B  X.  { (/) } )  X.  { 1o } ) )  i^i  ( ( C  X.  { 1o } )  X. 
{ 1o } ) )  =  (/) )  -> 
( ( A  +c  B )  +c  C
)  ~~  ( (
( A  X.  { (/)
} )  u.  (
( B  X.  { (/)
} )  X.  { 1o } ) )  u.  ( ( C  X.  { 1o } )  X. 
{ 1o } ) ) )
4823, 35, 46, 47syl3anc 1187 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  +c  B )  +c  C
)  ~~  ( (
( A  X.  { (/)
} )  u.  (
( B  X.  { (/)
} )  X.  { 1o } ) )  u.  ( ( C  X.  { 1o } )  X. 
{ 1o } ) ) )
49 ovex 5735 . . . . 5  |-  ( B  +c  C )  e. 
_V
50 cdaval 7680 . . . . 5  |-  ( ( A  e.  V  /\  ( B  +c  C
)  e.  _V )  ->  ( A  +c  ( B  +c  C ) )  =  ( ( A  X.  { (/) } )  u.  ( ( B  +c  C )  X. 
{ 1o } ) ) )
5149, 50mpan2 655 . . . 4  |-  ( A  e.  V  ->  ( A  +c  ( B  +c  C ) )  =  ( ( A  X.  { (/) } )  u.  ( ( B  +c  C )  X.  { 1o } ) ) )
52 cdaval 7680 . . . . . . . 8  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
5352xpeq1d 4619 . . . . . . 7  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( ( B  +c  C )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) )  X. 
{ 1o } ) )
54 xpundir 4649 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  u.  ( C  X.  { 1o }
) )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  X. 
{ 1o } )  u.  ( ( C  X.  { 1o }
)  X.  { 1o } ) )
5553, 54syl6eq 2301 . . . . . 6  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( ( B  +c  C )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  X. 
{ 1o } )  u.  ( ( C  X.  { 1o }
)  X.  { 1o } ) ) )
5655uneq2d 3239 . . . . 5  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( ( A  X.  { (/) } )  u.  ( ( B  +c  C )  X.  { 1o } ) )  =  ( ( A  X.  { (/) } )  u.  ( ( ( B  X.  { (/) } )  X.  { 1o }
)  u.  ( ( C  X.  { 1o } )  X.  { 1o } ) ) ) )
57 unass 3242 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  (
( B  X.  { (/)
} )  X.  { 1o } ) )  u.  ( ( C  X.  { 1o } )  X. 
{ 1o } ) )  =  ( ( A  X.  { (/) } )  u.  ( ( ( B  X.  { (/)
} )  X.  { 1o } )  u.  (
( C  X.  { 1o } )  X.  { 1o } ) ) )
5856, 57syl6eqr 2303 . . . 4  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( ( A  X.  { (/) } )  u.  ( ( B  +c  C )  X.  { 1o } ) )  =  ( ( ( A  X.  { (/) } )  u.  ( ( B  X.  { (/) } )  X.  { 1o }
) )  u.  (
( C  X.  { 1o } )  X.  { 1o } ) ) )
5951, 58sylan9eq 2305 . . 3  |-  ( ( A  e.  V  /\  ( B  e.  W  /\  C  e.  X
) )  ->  ( A  +c  ( B  +c  C ) )  =  ( ( ( A  X.  { (/) } )  u.  ( ( B  X.  { (/) } )  X.  { 1o }
) )  u.  (
( C  X.  { 1o } )  X.  { 1o } ) ) )
60593impb 1152 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  +c  ( B  +c  C ) )  =  ( ( ( A  X.  { (/) } )  u.  ( ( B  X.  { (/) } )  X.  { 1o } ) )  u.  ( ( C  X.  { 1o } )  X. 
{ 1o } ) ) )
6148, 60breqtrrd 3946 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  +c  B )  +c  C
)  ~~  ( A  +c  ( B  +c  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   _Vcvv 2727    u. cun 3076    i^i cin 3077   (/)c0 3362   {csn 3544   class class class wbr 3920   Oncon0 4285    X. cxp 4578  (class class class)co 5710   1oc1o 6358    ~~ cen 6746    +c ccda 7677
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1o 6365  df-er 6546  df-en 6750  df-cda 7678
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