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Theorem cdaassen 8022
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 4303 . . . . . 6  |-  (/)  e.  _V
3 xpsneng 7156 . . . . . 6  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
41, 2, 3sylancl 644 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  { (/)
} )  ~~  A
)
54ensymd 7121 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  ~~  ( A  X.  { (/) } ) )
6 simp2 958 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  e.  W )
7 snex 4369 . . . . . . . 8  |-  { (/) }  e.  _V
8 xpexg 4952 . . . . . . . 8  |-  ( ( B  e.  W  /\  {
(/) }  e.  _V )  ->  ( B  X.  { (/) } )  e. 
_V )
96, 7, 8sylancl 644 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  e.  _V )
10 1on 6694 . . . . . . 7  |-  1o  e.  On
11 xpsneng 7156 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  e.  _V  /\  1o  e.  On )  ->  ( ( B  X.  { (/) } )  X.  { 1o }
)  ~~  ( B  X.  { (/) } ) )
129, 10, 11sylancl 644 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( B  X.  { (/) } )  X. 
{ 1o } ) 
~~  ( B  X.  { (/) } ) )
13 xpsneng 7156 . . . . . . 7  |-  ( ( B  e.  W  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
146, 2, 13sylancl 644 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  ~~  B
)
15 entr 7122 . . . . . 6  |-  ( ( ( ( B  X.  { (/) } )  X. 
{ 1o } ) 
~~  ( B  X.  { (/) } )  /\  ( B  X.  { (/) } )  ~~  B )  ->  ( ( B  X.  { (/) } )  X.  { 1o }
)  ~~  B )
1612, 14, 15syl2anc 643 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( B  X.  { (/) } )  X. 
{ 1o } ) 
~~  B )
1716ensymd 7121 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  ~~  ( ( B  X.  { (/) } )  X.  { 1o } ) )
18 xp01disj 6703 . . . . 5  |-  ( ( A  X.  { (/) } )  i^i  ( ( B  X.  { (/) } )  X.  { 1o } ) )  =  (/)
1918a1i 11 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  X.  { (/) } )  i^i  ( ( B  X.  { (/) } )  X. 
{ 1o } ) )  =  (/) )
20 cdaenun 8014 . . . 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 } ) ) )
215, 17, 19, 20syl3anc 1184 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  +c  B
)  ~~  ( ( A  X.  { (/) } )  u.  ( ( B  X.  { (/) } )  X.  { 1o }
) ) )
22 simp3 959 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  e.  X )
23 snex 4369 . . . . . . 7  |-  { 1o }  e.  _V
24 xpexg 4952 . . . . . . 7  |-  ( ( C  e.  X  /\  { 1o }  e.  _V )  ->  ( C  X.  { 1o } )  e. 
_V )
2522, 23, 24sylancl 644 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  e.  _V )
26 xpsneng 7156 . . . . . 6  |-  ( ( ( C  X.  { 1o } )  e.  _V  /\  1o  e.  On )  ->  ( ( C  X.  { 1o }
)  X.  { 1o } )  ~~  ( C  X.  { 1o }
) )
2725, 10, 26sylancl 644 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( C  X.  { 1o } )  X. 
{ 1o } ) 
~~  ( C  X.  { 1o } ) )
28 xpsneng 7156 . . . . . 6  |-  ( ( C  e.  X  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
2922, 10, 28sylancl 644 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  ~~  C
)
30 entr 7122 . . . . 5  |-  ( ( ( ( C  X.  { 1o } )  X. 
{ 1o } ) 
~~  ( C  X.  { 1o } )  /\  ( C  X.  { 1o } )  ~~  C
)  ->  ( ( C  X.  { 1o }
)  X.  { 1o } )  ~~  C
)
3127, 29, 30syl2anc 643 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( C  X.  { 1o } )  X. 
{ 1o } ) 
~~  C )
3231ensymd 7121 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  ~~  ( ( C  X.  { 1o } )  X.  { 1o } ) )
33 indir 3553 . . . . 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 } ) ) )
34 xp01disj 6703 . . . . . 6  |-  ( ( A  X.  { (/) } )  i^i  ( ( C  X.  { 1o } )  X.  { 1o } ) )  =  (/)
35 xp01disj 6703 . . . . . . . 8  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
3635xpeq1i 4861 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  i^i  ( C  X.  { 1o }
) )  X.  { 1o } )  =  (
(/)  X.  { 1o } )
37 xpindir 4972 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  i^i  ( C  X.  { 1o }
) )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  X. 
{ 1o } )  i^i  ( ( C  X.  { 1o }
)  X.  { 1o } ) )
38 xp0r 4919 . . . . . . 7  |-  ( (/)  X. 
{ 1o } )  =  (/)
3936, 37, 383eqtr3i 2436 . . . . . 6  |-  ( ( ( B  X.  { (/)
} )  X.  { 1o } )  i^i  (
( C  X.  { 1o } )  X.  { 1o } ) )  =  (/)
4034, 39uneq12i 3463 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  i^i  (
( C  X.  { 1o } )  X.  { 1o } ) )  u.  ( ( ( B  X.  { (/) } )  X.  { 1o }
)  i^i  ( ( C  X.  { 1o }
)  X.  { 1o } ) ) )  =  ( (/)  u.  (/) )
41 un0 3616 . . . . 5  |-  ( (/)  u.  (/) )  =  (/)
4233, 40, 413eqtri 2432 . . . 4  |-  ( ( ( A  X.  { (/)
} )  u.  (
( B  X.  { (/)
} )  X.  { 1o } ) )  i^i  ( ( C  X.  { 1o } )  X. 
{ 1o } ) )  =  (/)
4342a1i 11 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( ( A  X.  { (/) } )  u.  ( ( B  X.  { (/) } )  X.  { 1o }
) )  i^i  (
( C  X.  { 1o } )  X.  { 1o } ) )  =  (/) )
44 cdaenun 8014 . . 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 } ) ) )
4521, 32, 43, 44syl3anc 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 } ) ) )
46 ovex 6069 . . . . 5  |-  ( B  +c  C )  e. 
_V
47 cdaval 8010 . . . . 5  |-  ( ( A  e.  V  /\  ( B  +c  C
)  e.  _V )  ->  ( A  +c  ( B  +c  C ) )  =  ( ( A  X.  { (/) } )  u.  ( ( B  +c  C )  X. 
{ 1o } ) ) )
4846, 47mpan2 653 . . . 4  |-  ( A  e.  V  ->  ( A  +c  ( B  +c  C ) )  =  ( ( A  X.  { (/) } )  u.  ( ( B  +c  C )  X.  { 1o } ) ) )
49 cdaval 8010 . . . . . . . 8  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
5049xpeq1d 4864 . . . . . . 7  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( ( B  +c  C )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) )  X. 
{ 1o } ) )
51 xpundir 4894 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  u.  ( C  X.  { 1o }
) )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  X. 
{ 1o } )  u.  ( ( C  X.  { 1o }
)  X.  { 1o } ) )
5250, 51syl6eq 2456 . . . . . 6  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( ( B  +c  C )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  X. 
{ 1o } )  u.  ( ( C  X.  { 1o }
)  X.  { 1o } ) ) )
5352uneq2d 3465 . . . . 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 } ) ) ) )
54 unass 3468 . . . . 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 } ) ) )
5553, 54syl6eqr 2458 . . . 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 } ) ) )
5648, 55sylan9eq 2460 . . 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 } ) ) )
57563impb 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 } ) ) )
5845, 57breqtrrd 4202 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2920    u. cun 3282    i^i cin 3283   (/)c0 3592   {csn 3778   class class class wbr 4176   Oncon0 4545    X. cxp 4839  (class class class)co 6044   1oc1o 6680    ~~ cen 7069    +c ccda 8007
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-suc 4551  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1o 6687  df-er 6868  df-en 7073  df-cda 8008
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