MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cdaassen Unicode version

Theorem cdaassen 7898
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 4231 . . . . . 6  |-  (/)  e.  _V
3 xpsneng 7035 . . . . . 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 6998 . . . . 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 4297 . . . . . . . 8  |-  { (/) }  e.  _V
9 xpexg 4882 . . . . . . . 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 6573 . . . . . . 7  |-  1o  e.  On
12 xpsneng 7035 . . . . . . 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 7035 . . . . . . 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 7001 . . . . . 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 6998 . . . . 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 6582 . . . . 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 7890 . . . 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 4297 . . . . . . 7  |-  { 1o }  e.  _V
26 xpexg 4882 . . . . . . 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 7035 . . . . . 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 7035 . . . . . 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 7001 . . . . 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 6998 . . . 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 3493 . . . . 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 6582 . . . . . 6  |-  ( ( A  X.  { (/) } )  i^i  ( ( C  X.  { 1o } )  X.  { 1o } ) )  =  (/)
38 xp01disj 6582 . . . . . . . 8  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
3938xpeq1i 4791 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  i^i  ( C  X.  { 1o }
) )  X.  { 1o } )  =  (
(/)  X.  { 1o } )
40 xpindir 4902 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  i^i  ( C  X.  { 1o }
) )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  X. 
{ 1o } )  i^i  ( ( C  X.  { 1o }
)  X.  { 1o } ) )
41 xp0r 4850 . . . . . . 7  |-  ( (/)  X. 
{ 1o } )  =  (/)
4239, 40, 413eqtr3i 2386 . . . . . 6  |-  ( ( ( B  X.  { (/)
} )  X.  { 1o } )  i^i  (
( C  X.  { 1o } )  X.  { 1o } ) )  =  (/)
4337, 42uneq12i 3403 . . . . 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 3555 . . . . 5  |-  ( (/)  u.  (/) )  =  (/)
4536, 43, 443eqtri 2382 . . . 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 7890 . . 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 5970 . . . . 5  |-  ( B  +c  C )  e. 
_V
50 cdaval 7886 . . . . 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 7886 . . . . . . . 8  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
5352xpeq1d 4794 . . . . . . 7  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( ( B  +c  C )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) )  X. 
{ 1o } ) )
54 xpundir 4824 . . . . . . 7  |-  ( ( ( B  X.  { (/)
} )  u.  ( C  X.  { 1o }
) )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  X. 
{ 1o } )  u.  ( ( C  X.  { 1o }
)  X.  { 1o } ) )
5553, 54syl6eq 2406 . . . . . 6  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( ( B  +c  C )  X.  { 1o } )  =  ( ( ( B  X.  { (/) } )  X. 
{ 1o } )  u.  ( ( C  X.  { 1o }
)  X.  { 1o } ) ) )
5655uneq2d 3405 . . . . 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 3408 . . . . 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 2408 . . . 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 2410 . . 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 4130 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 1642    e. wcel 1710   _Vcvv 2864    u. cun 3226    i^i cin 3227   (/)c0 3531   {csn 3716   class class class wbr 4104   Oncon0 4474    X. cxp 4769  (class class class)co 5945   1oc1o 6559    ~~ cen 6948    +c ccda 7883
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-suc 4480  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1o 6566  df-er 6747  df-en 6952  df-cda 7884
  Copyright terms: Public domain W3C validator