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Theorem cdacomen 7740
Description: Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
cdacomen  |-  ( A  +c  B )  ~~  ( B  +c  A
)

Proof of Theorem cdacomen
StepHypRef Expression
1 1on 6419 . . . . 5  |-  1o  e.  On
2 xpsneng 6880 . . . . 5  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  X.  { 1o } )  ~~  A
)
31, 2mpan2 655 . . . 4  |-  ( A  e.  _V  ->  ( A  X.  { 1o }
)  ~~  A )
4 0ex 4090 . . . . 5  |-  (/)  e.  _V
5 xpsneng 6880 . . . . 5  |-  ( ( B  e.  _V  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
64, 5mpan2 655 . . . 4  |-  ( B  e.  _V  ->  ( B  X.  { (/) } ) 
~~  B )
7 ensym 6843 . . . . 5  |-  ( ( A  X.  { 1o } )  ~~  A  ->  A  ~~  ( A  X.  { 1o }
) )
8 ensym 6843 . . . . 5  |-  ( ( B  X.  { (/) } )  ~~  B  ->  B  ~~  ( B  X.  { (/) } ) )
9 incom 3303 . . . . . . 7  |-  ( ( A  X.  { 1o } )  i^i  ( B  X.  { (/) } ) )  =  ( ( B  X.  { (/) } )  i^i  ( A  X.  { 1o }
) )
10 xp01disj 6428 . . . . . . 7  |-  ( ( B  X.  { (/) } )  i^i  ( A  X.  { 1o }
) )  =  (/)
119, 10eqtri 2276 . . . . . 6  |-  ( ( A  X.  { 1o } )  i^i  ( B  X.  { (/) } ) )  =  (/)
12 cdaenun 7733 . . . . . 6  |-  ( ( A  ~~  ( A  X.  { 1o }
)  /\  B  ~~  ( B  X.  { (/) } )  /\  ( ( A  X.  { 1o } )  i^i  ( B  X.  { (/) } ) )  =  (/) )  -> 
( A  +c  B
)  ~~  ( ( A  X.  { 1o }
)  u.  ( B  X.  { (/) } ) ) )
1311, 12mp3an3 1271 . . . . 5  |-  ( ( A  ~~  ( A  X.  { 1o }
)  /\  B  ~~  ( B  X.  { (/) } ) )  ->  ( A  +c  B )  ~~  ( ( A  X.  { 1o } )  u.  ( B  X.  { (/)
} ) ) )
147, 8, 13syl2an 465 . . . 4  |-  ( ( ( A  X.  { 1o } )  ~~  A  /\  ( B  X.  { (/)
} )  ~~  B
)  ->  ( A  +c  B )  ~~  (
( A  X.  { 1o } )  u.  ( B  X.  { (/) } ) ) )
153, 6, 14syl2an 465 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  ~~  ( ( A  X.  { 1o }
)  u.  ( B  X.  { (/) } ) ) )
16 cdaval 7729 . . . . 5  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( B  +c  A
)  =  ( ( B  X.  { (/) } )  u.  ( A  X.  { 1o }
) ) )
1716ancoms 441 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( B  +c  A
)  =  ( ( B  X.  { (/) } )  u.  ( A  X.  { 1o }
) ) )
18 uncom 3261 . . . 4  |-  ( ( B  X.  { (/) } )  u.  ( A  X.  { 1o }
) )  =  ( ( A  X.  { 1o } )  u.  ( B  X.  { (/) } ) )
1917, 18syl6eq 2304 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( B  +c  A
)  =  ( ( A  X.  { 1o } )  u.  ( B  X.  { (/) } ) ) )
2015, 19breqtrrd 3989 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  ~~  ( B  +c  A ) )
214enref 6827 . . . 4  |-  (/)  ~~  (/)
2221a1i 12 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
(/)  ~~  (/) )
23 cdafn 7728 . . . . 5  |-  +c  Fn  ( _V  X.  _V )
24 fndm 5246 . . . . 5  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
2523, 24ax-mp 10 . . . 4  |-  dom  +c  =  ( _V  X.  _V )
2625ndmov 5903 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  (/) )
27 ancom 439 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( B  e.  _V  /\  A  e.  _V )
)
2825ndmov 5903 . . . 4  |-  ( -.  ( B  e.  _V  /\  A  e.  _V )  ->  ( B  +c  A
)  =  (/) )
2927, 28sylnbi 299 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( B  +c  A
)  =  (/) )
3022, 26, 293brtr4d 3993 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  ~~  ( B  +c  A ) )
3120, 30pm2.61i 158 1  |-  ( A  +c  B )  ~~  ( B  +c  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2740    u. cun 3092    i^i cin 3093   (/)c0 3397   {csn 3581   class class class wbr 3963   Oncon0 4329    X. cxp 4624   dom cdm 4626    Fn wfn 4633  (class class class)co 5757   1oc1o 6405    ~~ cen 6793    +c ccda 7726
This theorem is referenced by:  cdadom2  7746  cdalepw  7755  infcda  7767  alephadd  8132  gchdomtri  8184  pwxpndom  8221  gchhar  8226  gchpwdom  8229
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 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-suc 4335  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-1o 6412  df-er 6593  df-en 6797  df-cda 7727
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