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Theorem cdacomen 8021
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 6694 . . . . 5  |-  1o  e.  On
2 xpsneng 7156 . . . . 5  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  X.  { 1o } )  ~~  A
)
31, 2mpan2 653 . . . 4  |-  ( A  e.  _V  ->  ( A  X.  { 1o }
)  ~~  A )
4 0ex 4303 . . . . 5  |-  (/)  e.  _V
5 xpsneng 7156 . . . . 5  |-  ( ( B  e.  _V  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
64, 5mpan2 653 . . . 4  |-  ( B  e.  _V  ->  ( B  X.  { (/) } ) 
~~  B )
7 ensym 7119 . . . . 5  |-  ( ( A  X.  { 1o } )  ~~  A  ->  A  ~~  ( A  X.  { 1o }
) )
8 ensym 7119 . . . . 5  |-  ( ( B  X.  { (/) } )  ~~  B  ->  B  ~~  ( B  X.  { (/) } ) )
9 incom 3497 . . . . . . 7  |-  ( ( A  X.  { 1o } )  i^i  ( B  X.  { (/) } ) )  =  ( ( B  X.  { (/) } )  i^i  ( A  X.  { 1o }
) )
10 xp01disj 6703 . . . . . . 7  |-  ( ( B  X.  { (/) } )  i^i  ( A  X.  { 1o }
) )  =  (/)
119, 10eqtri 2428 . . . . . 6  |-  ( ( A  X.  { 1o } )  i^i  ( B  X.  { (/) } ) )  =  (/)
12 cdaenun 8014 . . . . . 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 464 . . . 4  |-  ( ( ( A  X.  { 1o } )  ~~  A  /\  ( B  X.  { (/)
} )  ~~  B
)  ->  ( A  +c  B )  ~~  (
( A  X.  { 1o } )  u.  ( B  X.  { (/) } ) ) )
153, 6, 14syl2an 464 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  ~~  ( ( A  X.  { 1o }
)  u.  ( B  X.  { (/) } ) ) )
16 cdaval 8010 . . . . 5  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( B  +c  A
)  =  ( ( B  X.  { (/) } )  u.  ( A  X.  { 1o }
) ) )
1716ancoms 440 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( B  +c  A
)  =  ( ( B  X.  { (/) } )  u.  ( A  X.  { 1o }
) ) )
18 uncom 3455 . . . 4  |-  ( ( B  X.  { (/) } )  u.  ( A  X.  { 1o }
) )  =  ( ( A  X.  { 1o } )  u.  ( B  X.  { (/) } ) )
1917, 18syl6eq 2456 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( B  +c  A
)  =  ( ( A  X.  { 1o } )  u.  ( B  X.  { (/) } ) ) )
2015, 19breqtrrd 4202 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  ~~  ( B  +c  A ) )
214enref 7103 . . . 4  |-  (/)  ~~  (/)
2221a1i 11 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
(/)  ~~  (/) )
23 cdafn 8009 . . . . 5  |-  +c  Fn  ( _V  X.  _V )
24 fndm 5507 . . . . 5  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
2523, 24ax-mp 8 . . . 4  |-  dom  +c  =  ( _V  X.  _V )
2625ndmov 6194 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  (/) )
27 ancom 438 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( B  e.  _V  /\  A  e.  _V )
)
2825ndmov 6194 . . . 4  |-  ( -.  ( B  e.  _V  /\  A  e.  _V )  ->  ( B  +c  A
)  =  (/) )
2927, 28sylnbi 298 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( B  +c  A
)  =  (/) )
3022, 26, 293brtr4d 4206 . 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 3    /\ wa 359    = 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   dom cdm 4841    Fn wfn 5412  (class class class)co 6044   1oc1o 6680    ~~ cen 7069    +c ccda 8007
This theorem is referenced by:  cdadom2  8027  cdalepw  8036  infcda  8048  alephadd  8412  gchdomtri  8464  pwxpndom  8501  gchhar  8506  gchpwdom  8509
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-csb 3216  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-iun 4059  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-1st 6312  df-2nd 6313  df-1o 6687  df-er 6868  df-en 7073  df-cda 8008
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