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Theorem cdaval 8618
Description: Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258. For cardinal arithmetic, we follow Mendelson. Rather than defining operations restricted to cardinal numbers, we use this disjoint union operation for addition, while Cartesian product and set exponentiation stand in for cardinal multiplication and exponentiation. Equinumerosity and dominance serve the roles of equality and ordering. If we wanted to, we could easily convert our theorems to actual cardinal number operations via carden 8994, carddom 8997, and cardsdom 8998. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cdaval  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )

Proof of Theorem cdaval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3040 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 3040 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 p0ex 4588 . . . . . 6  |-  { (/) }  e.  _V
4 xpexg 6612 . . . . . 6  |-  ( ( A  e.  _V  /\  {
(/) }  e.  _V )  ->  ( A  X.  { (/) } )  e. 
_V )
53, 4mpan2 685 . . . . 5  |-  ( A  e.  _V  ->  ( A  X.  { (/) } )  e.  _V )
6 snex 4641 . . . . . 6  |-  { 1o }  e.  _V
7 xpexg 6612 . . . . . 6  |-  ( ( B  e.  _V  /\  { 1o }  e.  _V )  ->  ( B  X.  { 1o } )  e. 
_V )
86, 7mpan2 685 . . . . 5  |-  ( B  e.  _V  ->  ( B  X.  { 1o }
)  e.  _V )
95, 8anim12i 576 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( A  X.  { (/) } )  e. 
_V  /\  ( B  X.  { 1o } )  e.  _V ) )
10 unexb 6610 . . . 4  |-  ( ( ( A  X.  { (/)
} )  e.  _V  /\  ( B  X.  { 1o } )  e.  _V ) 
<->  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) )  e. 
_V )
119, 10sylib 201 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) )  e. 
_V )
12 xpeq1 4853 . . . . 5  |-  ( x  =  A  ->  (
x  X.  { (/) } )  =  ( A  X.  { (/) } ) )
1312uneq1d 3578 . . . 4  |-  ( x  =  A  ->  (
( x  X.  { (/)
} )  u.  (
y  X.  { 1o } ) )  =  ( ( A  X.  { (/) } )  u.  ( y  X.  { 1o } ) ) )
14 xpeq1 4853 . . . . 5  |-  ( y  =  B  ->  (
y  X.  { 1o } )  =  ( B  X.  { 1o } ) )
1514uneq2d 3579 . . . 4  |-  ( y  =  B  ->  (
( A  X.  { (/)
} )  u.  (
y  X.  { 1o } ) )  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
16 df-cda 8616 . . . 4  |-  +c  =  ( x  e.  _V ,  y  e.  _V  |->  ( ( x  X.  { (/) } )  u.  ( y  X.  { 1o } ) ) )
1713, 15, 16ovmpt2g 6450 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  (
( A  X.  { (/)
} )  u.  ( B  X.  { 1o }
) )  e.  _V )  ->  ( A  +c  B )  =  ( ( A  X.  { (/)
} )  u.  ( B  X.  { 1o }
) ) )
1811, 17mpd3an3 1391 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
191, 2, 18syl2an 485 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031    u. cun 3388   (/)c0 3722   {csn 3959    X. cxp 4837  (class class class)co 6308   1oc1o 7193    +c ccda 8615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-cda 8616
This theorem is referenced by:  uncdadom  8619  cdaun  8620  cdaen  8621  cda1dif  8624  pm110.643  8625  xp2cda  8628  cdacomen  8629  cdaassen  8630  xpcdaen  8631  mapcdaen  8632  cdadom1  8634  cdaxpdom  8637  cdafi  8638  cdainf  8640  infcda1  8641  pwcdadom  8664  isfin4-3  8763  alephadd  9020  canthp1lem2  9096  xpsc  15541
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