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Theorem cdaassen 7803
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 957 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  e.  V )
2 0ex 4151 . . . . . 6  |-  (/)  e.  _V
3 xpsneng 6942 . . . . . 6  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
41, 2, 3sylancl 645 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  { (/)
} )  ~~  A
)
5 ensym 6905 . . . . 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 958 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  e.  W )
8 snex 4215 . . . . . . . 8  |-  { (/) }  e.  _V
9 xpexg 4799 . . . . . . . 8  |-  ( ( B  e.  W  /\  {
(/) }  e.  _V )  ->  ( B  X.  { (/) } )  e. 
_V )
107, 8, 9sylancl 645 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  e.  _V )
11 1on 6481 . . . . . . 7  |-  1o  e.  On
12 xpsneng 6942 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  e.  _V  /\  1o  e.  On )  ->  ( ( B  X.  { (/) } )  X.  { 1o }
)  ~~  ( B  X.  { (/) } ) )
1310, 11, 12sylancl 645 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( B  X.  { (/) } )  X. 
{ 1o } ) 
~~  ( B  X.  { (/) } ) )
14 xpsneng 6942 . . . . . . 7  |-  ( ( B  e.  W  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
157, 2, 14sylancl 645 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  ~~  B
)
16 entr 6908 . . . . . 6  |-  ( ( ( ( B  X.  { (/) } )  X. 
{ 1o } ) 
~~  ( B  X.  { (/) } )  /\  ( B  X.  { (/) } )  ~~  B )  ->  ( ( B  X.  { (/) } )  X.  { 1o }
)  ~~  B )
1713, 15, 16syl2anc 644 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( B  X.  { (/) } )  X. 
{ 1o } ) 
~~  B )
18 ensym 6905 . . . . 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 6490 . . . . 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 7795 . . . 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 1184 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  +c  B
)  ~~  ( ( A  X.  { (/) } )  u.  ( ( B  X.  { (/) } )  X.  { 1o }
) ) )
24 simp3 959 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  e.  X )
25 snex 4215 . . . . . . 7  |-  { 1o }  e.  _V
26 xpexg 4799 . . . . . . 7  |-  ( ( C  e.  X  /\  { 1o }  e.  _V )  ->  ( C  X.  { 1o } )  e. 
_V )
2724, 25, 26sylancl 645 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  e.  _V )
28 xpsneng 6942 . . . . . 6  |-  ( ( ( C  X.  { 1o } )  e.  _V  /\  1o  e.  On )  ->  ( ( C  X.  { 1o }
)  X.  { 1o } )  ~~  ( C  X.  { 1o }
) )
2927, 11, 28sylancl 645 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( C  X.  { 1o } )  X. 
{ 1o } ) 
~~  ( C  X.  { 1o } ) )
30 xpsneng 6942 . . . . . 6  |-  ( ( C  e.  X  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
3124, 11, 30sylancl 645 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  ~~  C
)
32 entr 6908 . . . . 5  |-  ( ( ( ( C  X.  { 1o } )  X. 
{ 1o } ) 
~~  ( C  X.  { 1o } )  /\  ( C  X.  { 1o } )  ~~  C
)  ->  ( ( C  X.  { 1o }
)  X.  { 1o } )  ~~  C
)
3329, 31, 32syl2anc 644 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( C  X.  { 1o } )  X. 
{ 1o } ) 
~~  C )
34 ensym 6905 . . . 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 3418 . . . . 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 6490 . . . . . 6  |-  ( ( A  X.  { (/) } )  i^i  ( ( C  X.  { 1o } )  X.  { 1o } ) )  =  (/)
38 xp01disj 6490 . . . . . . . 8  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
3938xpeq1i 4708 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  i^i  ( C  X.  { 1o }
) )  X.  { 1o } )  =  (
(/)  X.  { 1o } )
40 xpindir 4819 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  i^i  ( C  X.  { 1o }
) )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  X. 
{ 1o } )  i^i  ( ( C  X.  { 1o }
)  X.  { 1o } ) )
41 xp0r 4767 . . . . . . 7  |-  ( (/)  X. 
{ 1o } )  =  (/)
4239, 40, 413eqtr3i 2312 . . . . . 6  |-  ( ( ( B  X.  { (/)
} )  X.  { 1o } )  i^i  (
( C  X.  { 1o } )  X.  { 1o } ) )  =  (/)
4337, 42uneq12i 3328 . . . . 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 3480 . . . . 5  |-  ( (/)  u.  (/) )  =  (/)
4536, 43, 443eqtri 2308 . . . 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 7795 . . 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 1184 . 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 5844 . . . . 5  |-  ( B  +c  C )  e. 
_V
50 cdaval 7791 . . . . 5  |-  ( ( A  e.  V  /\  ( B  +c  C
)  e.  _V )  ->  ( A  +c  ( B  +c  C ) )  =  ( ( A  X.  { (/) } )  u.  ( ( B  +c  C )  X. 
{ 1o } ) ) )
5149, 50mpan2 654 . . . 4  |-  ( A  e.  V  ->  ( A  +c  ( B  +c  C ) )  =  ( ( A  X.  { (/) } )  u.  ( ( B  +c  C )  X.  { 1o } ) ) )
52 cdaval 7791 . . . . . . . 8  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
5352xpeq1d 4711 . . . . . . 7  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( ( B  +c  C )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) )  X. 
{ 1o } ) )
54 xpundir 4741 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  u.  ( C  X.  { 1o }
) )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  X. 
{ 1o } )  u.  ( ( C  X.  { 1o }
)  X.  { 1o } ) )
5553, 54syl6eq 2332 . . . . . 6  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( ( B  +c  C )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  X. 
{ 1o } )  u.  ( ( C  X.  { 1o }
)  X.  { 1o } ) ) )
5655uneq2d 3330 . . . . 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 3333 . . . . 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 2334 . . . 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 2336 . . 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 1149 . 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 4050 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 936    = wceq 1624    e. wcel 1685   _Vcvv 2789    u. cun 3151    i^i cin 3152   (/)c0 3456   {csn 3641   class class class wbr 4024   Oncon0 4391    X. cxp 4686  (class class class)co 5819   1oc1o 6467    ~~ cen 6855    +c ccda 7788
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1o 6474  df-er 6655  df-en 6859  df-cda 7789
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