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Theorem cdaval 7796
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 cross 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 8173, carddom 8176, and cardsdom 8177. 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 2796 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2796 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 p0ex 4197 . . . . . 6  |-  { (/) }  e.  _V
4 xpexg 4800 . . . . . 6  |-  ( ( A  e.  _V  /\  {
(/) }  e.  _V )  ->  ( A  X.  { (/) } )  e. 
_V )
53, 4mpan2 652 . . . . 5  |-  ( A  e.  _V  ->  ( A  X.  { (/) } )  e.  _V )
6 snex 4216 . . . . . 6  |-  { 1o }  e.  _V
7 xpexg 4800 . . . . . 6  |-  ( ( B  e.  _V  /\  { 1o }  e.  _V )  ->  ( B  X.  { 1o } )  e. 
_V )
86, 7mpan2 652 . . . . 5  |-  ( B  e.  _V  ->  ( B  X.  { 1o }
)  e.  _V )
95, 8anim12i 549 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( A  X.  { (/) } )  e. 
_V  /\  ( B  X.  { 1o } )  e.  _V ) )
10 unexb 4520 . . . 4  |-  ( ( ( A  X.  { (/)
} )  e.  _V  /\  ( B  X.  { 1o } )  e.  _V ) 
<->  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) )  e. 
_V )
119, 10sylib 188 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) )  e. 
_V )
12 xpeq1 4703 . . . . 5  |-  ( x  =  A  ->  (
x  X.  { (/) } )  =  ( A  X.  { (/) } ) )
1312uneq1d 3328 . . . 4  |-  ( x  =  A  ->  (
( x  X.  { (/)
} )  u.  (
y  X.  { 1o } ) )  =  ( ( A  X.  { (/) } )  u.  ( y  X.  { 1o } ) ) )
14 xpeq1 4703 . . . . 5  |-  ( y  =  B  ->  (
y  X.  { 1o } )  =  ( B  X.  { 1o } ) )
1514uneq2d 3329 . . . 4  |-  ( y  =  B  ->  (
( A  X.  { (/)
} )  u.  (
y  X.  { 1o } ) )  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
16 df-cda 7794 . . . 4  |-  +c  =  ( x  e.  _V ,  y  e.  _V  |->  ( ( x  X.  { (/) } )  u.  ( y  X.  { 1o } ) ) )
1713, 15, 16ovmpt2g 5982 . . 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 1278 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
191, 2, 18syl2an 463 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150   (/)c0 3455   {csn 3640    X. cxp 4687  (class class class)co 5858   1oc1o 6472    +c ccda 7793
This theorem is referenced by:  uncdadom  7797  cdaun  7798  cdaen  7799  cda1dif  7802  pm110.643  7803  xp2cda  7806  cdacomen  7807  cdaassen  7808  xpcdaen  7809  mapcdaen  7810  cdadom1  7812  cdaxpdom  7815  cdafi  7816  cdainf  7818  infcda1  7819  pwcdadom  7842  isfin4-3  7941  alephadd  8199  canthp1lem2  8275  xpsc  13459
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-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-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-cda 7794
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