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Theorem cdacomen 7807
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 6486 . . . . 5  |-  1o  e.  On
2 xpsneng 6947 . . . . 5  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  X.  { 1o } )  ~~  A
)
31, 2mpan2 652 . . . 4  |-  ( A  e.  _V  ->  ( A  X.  { 1o }
)  ~~  A )
4 0ex 4150 . . . . 5  |-  (/)  e.  _V
5 xpsneng 6947 . . . . 5  |-  ( ( B  e.  _V  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
64, 5mpan2 652 . . . 4  |-  ( B  e.  _V  ->  ( B  X.  { (/) } ) 
~~  B )
7 ensym 6910 . . . . 5  |-  ( ( A  X.  { 1o } )  ~~  A  ->  A  ~~  ( A  X.  { 1o }
) )
8 ensym 6910 . . . . 5  |-  ( ( B  X.  { (/) } )  ~~  B  ->  B  ~~  ( B  X.  { (/) } ) )
9 incom 3361 . . . . . . 7  |-  ( ( A  X.  { 1o } )  i^i  ( B  X.  { (/) } ) )  =  ( ( B  X.  { (/) } )  i^i  ( A  X.  { 1o }
) )
10 xp01disj 6495 . . . . . . 7  |-  ( ( B  X.  { (/) } )  i^i  ( A  X.  { 1o }
) )  =  (/)
119, 10eqtri 2303 . . . . . 6  |-  ( ( A  X.  { 1o } )  i^i  ( B  X.  { (/) } ) )  =  (/)
12 cdaenun 7800 . . . . . 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 1266 . . . . 5  |-  ( ( A  ~~  ( A  X.  { 1o }
)  /\  B  ~~  ( B  X.  { (/) } ) )  ->  ( A  +c  B )  ~~  ( ( A  X.  { 1o } )  u.  ( B  X.  { (/)
} ) ) )
147, 8, 13syl2an 463 . . . 4  |-  ( ( ( A  X.  { 1o } )  ~~  A  /\  ( B  X.  { (/)
} )  ~~  B
)  ->  ( A  +c  B )  ~~  (
( A  X.  { 1o } )  u.  ( B  X.  { (/) } ) ) )
153, 6, 14syl2an 463 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  ~~  ( ( A  X.  { 1o }
)  u.  ( B  X.  { (/) } ) ) )
16 cdaval 7796 . . . . 5  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( B  +c  A
)  =  ( ( B  X.  { (/) } )  u.  ( A  X.  { 1o }
) ) )
1716ancoms 439 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( B  +c  A
)  =  ( ( B  X.  { (/) } )  u.  ( A  X.  { 1o }
) ) )
18 uncom 3319 . . . 4  |-  ( ( B  X.  { (/) } )  u.  ( A  X.  { 1o }
) )  =  ( ( A  X.  { 1o } )  u.  ( B  X.  { (/) } ) )
1917, 18syl6eq 2331 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( B  +c  A
)  =  ( ( A  X.  { 1o } )  u.  ( B  X.  { (/) } ) ) )
2015, 19breqtrrd 4049 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  ~~  ( B  +c  A ) )
214enref 6894 . . . 4  |-  (/)  ~~  (/)
2221a1i 10 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
(/)  ~~  (/) )
23 cdafn 7795 . . . . 5  |-  +c  Fn  ( _V  X.  _V )
24 fndm 5343 . . . . 5  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
2523, 24ax-mp 8 . . . 4  |-  dom  +c  =  ( _V  X.  _V )
2625ndmov 6004 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  (/) )
27 ancom 437 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( B  e.  _V  /\  A  e.  _V )
)
2825ndmov 6004 . . . 4  |-  ( -.  ( B  e.  _V  /\  A  e.  _V )  ->  ( B  +c  A
)  =  (/) )
2927, 28sylnbi 297 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( B  +c  A
)  =  (/) )
3022, 26, 293brtr4d 4053 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  ~~  ( B  +c  A ) )
3120, 30pm2.61i 156 1  |-  ( A  +c  B )  ~~  ( B  +c  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = 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   dom cdm 4689    Fn wfn 5250  (class class class)co 5858   1oc1o 6472    ~~ cen 6860    +c ccda 7793
This theorem is referenced by:  cdadom2  7813  cdalepw  7822  infcda  7834  alephadd  8199  gchdomtri  8251  pwxpndom  8288  gchhar  8293  gchpwdom  8296
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-csb 3082  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-iun 3907  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-1st 6122  df-2nd 6123  df-1o 6479  df-er 6660  df-en 6864  df-cda 7794
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