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Theorem cdacomen 7802
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 6481 . . . . 5  |-  1o  e.  On
2 xpsneng 6942 . . . . 5  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  X.  { 1o } )  ~~  A
)
31, 2mpan2 654 . . . 4  |-  ( A  e.  _V  ->  ( A  X.  { 1o }
)  ~~  A )
4 0ex 4151 . . . . 5  |-  (/)  e.  _V
5 xpsneng 6942 . . . . 5  |-  ( ( B  e.  _V  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
64, 5mpan2 654 . . . 4  |-  ( B  e.  _V  ->  ( B  X.  { (/) } ) 
~~  B )
7 ensym 6905 . . . . 5  |-  ( ( A  X.  { 1o } )  ~~  A  ->  A  ~~  ( A  X.  { 1o }
) )
8 ensym 6905 . . . . 5  |-  ( ( B  X.  { (/) } )  ~~  B  ->  B  ~~  ( B  X.  { (/) } ) )
9 incom 3362 . . . . . . 7  |-  ( ( A  X.  { 1o } )  i^i  ( B  X.  { (/) } ) )  =  ( ( B  X.  { (/) } )  i^i  ( A  X.  { 1o }
) )
10 xp01disj 6490 . . . . . . 7  |-  ( ( B  X.  { (/) } )  i^i  ( A  X.  { 1o }
) )  =  (/)
119, 10eqtri 2304 . . . . . 6  |-  ( ( A  X.  { 1o } )  i^i  ( B  X.  { (/) } ) )  =  (/)
12 cdaenun 7795 . . . . . 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 1268 . . . . 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 7791 . . . . 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 3320 . . . 4  |-  ( ( B  X.  { (/) } )  u.  ( A  X.  { 1o }
) )  =  ( ( A  X.  { 1o } )  u.  ( B  X.  { (/) } ) )
1917, 18syl6eq 2332 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( B  +c  A
)  =  ( ( A  X.  { 1o } )  u.  ( B  X.  { (/) } ) ) )
2015, 19breqtrrd 4050 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  ~~  ( B  +c  A ) )
214enref 6889 . . . 4  |-  (/)  ~~  (/)
2221a1i 12 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
(/)  ~~  (/) )
23 cdafn 7790 . . . . 5  |-  +c  Fn  ( _V  X.  _V )
24 fndm 5308 . . . . 5  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
2523, 24ax-mp 10 . . . 4  |-  dom  +c  =  ( _V  X.  _V )
2625ndmov 5965 . . 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 5965 . . . 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 4054 . 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 1624    e. wcel 1685   _Vcvv 2789    u. cun 3151    i^i cin 3152   (/)c0 3456   {csn 3641   class class class wbr 4024   Oncon0 4391    X. cxp 4686   dom cdm 4688    Fn wfn 5216  (class class class)co 5819   1oc1o 6467    ~~ cen 6855    +c ccda 7788
This theorem is referenced by:  cdadom2  7808  cdalepw  7817  infcda  7829  alephadd  8194  gchdomtri  8246  pwxpndom  8283  gchhar  8288  gchpwdom  8291
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-1o 6474  df-er 6655  df-en 6859  df-cda 7789
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