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Theorem cdaassen 7808
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 955 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  e.  V )
2 0ex 4150 . . . . . 6  |-  (/)  e.  _V
3 xpsneng 6947 . . . . . 6  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
41, 2, 3sylancl 643 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  { (/)
} )  ~~  A
)
5 ensym 6910 . . . . 5  |-  ( ( A  X.  { (/) } )  ~~  A  ->  A  ~~  ( A  X.  { (/) } ) )
64, 5syl 15 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  ~~  ( A  X.  { (/) } ) )
7 simp2 956 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  e.  W )
8 snex 4216 . . . . . . . 8  |-  { (/) }  e.  _V
9 xpexg 4800 . . . . . . . 8  |-  ( ( B  e.  W  /\  {
(/) }  e.  _V )  ->  ( B  X.  { (/) } )  e. 
_V )
107, 8, 9sylancl 643 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  e.  _V )
11 1on 6486 . . . . . . 7  |-  1o  e.  On
12 xpsneng 6947 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  e.  _V  /\  1o  e.  On )  ->  ( ( B  X.  { (/) } )  X.  { 1o }
)  ~~  ( B  X.  { (/) } ) )
1310, 11, 12sylancl 643 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( B  X.  { (/) } )  X. 
{ 1o } ) 
~~  ( B  X.  { (/) } ) )
14 xpsneng 6947 . . . . . . 7  |-  ( ( B  e.  W  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
157, 2, 14sylancl 643 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  ~~  B
)
16 entr 6913 . . . . . 6  |-  ( ( ( ( B  X.  { (/) } )  X. 
{ 1o } ) 
~~  ( B  X.  { (/) } )  /\  ( B  X.  { (/) } )  ~~  B )  ->  ( ( B  X.  { (/) } )  X.  { 1o }
)  ~~  B )
1713, 15, 16syl2anc 642 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( B  X.  { (/) } )  X. 
{ 1o } ) 
~~  B )
18 ensym 6910 . . . . 5  |-  ( ( ( B  X.  { (/)
} )  X.  { 1o } )  ~~  B  ->  B  ~~  ( ( B  X.  { (/) } )  X.  { 1o } ) )
1917, 18syl 15 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  ~~  ( ( B  X.  { (/) } )  X.  { 1o } ) )
20 xp01disj 6495 . . . . 5  |-  ( ( A  X.  { (/) } )  i^i  ( ( B  X.  { (/) } )  X.  { 1o } ) )  =  (/)
2120a1i 10 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  X.  { (/) } )  i^i  ( ( B  X.  { (/) } )  X. 
{ 1o } ) )  =  (/) )
22 cdaenun 7800 . . . 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 1182 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  +c  B
)  ~~  ( ( A  X.  { (/) } )  u.  ( ( B  X.  { (/) } )  X.  { 1o }
) ) )
24 simp3 957 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  e.  X )
25 snex 4216 . . . . . . 7  |-  { 1o }  e.  _V
26 xpexg 4800 . . . . . . 7  |-  ( ( C  e.  X  /\  { 1o }  e.  _V )  ->  ( C  X.  { 1o } )  e. 
_V )
2724, 25, 26sylancl 643 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  e.  _V )
28 xpsneng 6947 . . . . . 6  |-  ( ( ( C  X.  { 1o } )  e.  _V  /\  1o  e.  On )  ->  ( ( C  X.  { 1o }
)  X.  { 1o } )  ~~  ( C  X.  { 1o }
) )
2927, 11, 28sylancl 643 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( C  X.  { 1o } )  X. 
{ 1o } ) 
~~  ( C  X.  { 1o } ) )
30 xpsneng 6947 . . . . . 6  |-  ( ( C  e.  X  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
3124, 11, 30sylancl 643 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  ~~  C
)
32 entr 6913 . . . . 5  |-  ( ( ( ( C  X.  { 1o } )  X. 
{ 1o } ) 
~~  ( C  X.  { 1o } )  /\  ( C  X.  { 1o } )  ~~  C
)  ->  ( ( C  X.  { 1o }
)  X.  { 1o } )  ~~  C
)
3329, 31, 32syl2anc 642 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( C  X.  { 1o } )  X. 
{ 1o } ) 
~~  C )
34 ensym 6910 . . . 4  |-  ( ( ( C  X.  { 1o } )  X.  { 1o } )  ~~  C  ->  C  ~~  ( ( C  X.  { 1o } )  X.  { 1o } ) )
3533, 34syl 15 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  ~~  ( ( C  X.  { 1o } )  X.  { 1o } ) )
36 indir 3417 . . . . 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 6495 . . . . . 6  |-  ( ( A  X.  { (/) } )  i^i  ( ( C  X.  { 1o } )  X.  { 1o } ) )  =  (/)
38 xp01disj 6495 . . . . . . . 8  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
3938xpeq1i 4709 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  i^i  ( C  X.  { 1o }
) )  X.  { 1o } )  =  (
(/)  X.  { 1o } )
40 xpindir 4820 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  i^i  ( C  X.  { 1o }
) )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  X. 
{ 1o } )  i^i  ( ( C  X.  { 1o }
)  X.  { 1o } ) )
41 xp0r 4768 . . . . . . 7  |-  ( (/)  X. 
{ 1o } )  =  (/)
4239, 40, 413eqtr3i 2311 . . . . . 6  |-  ( ( ( B  X.  { (/)
} )  X.  { 1o } )  i^i  (
( C  X.  { 1o } )  X.  { 1o } ) )  =  (/)
4337, 42uneq12i 3327 . . . . 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 3479 . . . . 5  |-  ( (/)  u.  (/) )  =  (/)
4536, 43, 443eqtri 2307 . . . 4  |-  ( ( ( A  X.  { (/)
} )  u.  (
( B  X.  { (/)
} )  X.  { 1o } ) )  i^i  ( ( C  X.  { 1o } )  X. 
{ 1o } ) )  =  (/)
4645a1i 10 . . 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 7800 . . 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 1182 . 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 5883 . . . . 5  |-  ( B  +c  C )  e. 
_V
50 cdaval 7796 . . . . 5  |-  ( ( A  e.  V  /\  ( B  +c  C
)  e.  _V )  ->  ( A  +c  ( B  +c  C ) )  =  ( ( A  X.  { (/) } )  u.  ( ( B  +c  C )  X. 
{ 1o } ) ) )
5149, 50mpan2 652 . . . 4  |-  ( A  e.  V  ->  ( A  +c  ( B  +c  C ) )  =  ( ( A  X.  { (/) } )  u.  ( ( B  +c  C )  X.  { 1o } ) ) )
52 cdaval 7796 . . . . . . . 8  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
5352xpeq1d 4712 . . . . . . 7  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( ( B  +c  C )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) )  X. 
{ 1o } ) )
54 xpundir 4742 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  u.  ( C  X.  { 1o }
) )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  X. 
{ 1o } )  u.  ( ( C  X.  { 1o }
)  X.  { 1o } ) )
5553, 54syl6eq 2331 . . . . . 6  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( ( B  +c  C )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  X. 
{ 1o } )  u.  ( ( C  X.  { 1o }
)  X.  { 1o } ) ) )
5655uneq2d 3329 . . . . 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 3332 . . . . 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 2333 . . . 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 2335 . . 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 1147 . 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 4049 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 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640   class class class wbr 4023   Oncon0 4392    X. cxp 4687  (class class class)co 5858   1oc1o 6472    ~~ cen 6860    +c ccda 7793
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1o 6479  df-er 6660  df-en 6864  df-cda 7794
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