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Theorem cdaval 7791
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 8168, carddom 8171, and cardsdom 8172. 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 }
) ) )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem cdaval
StepHypRef Expression
1 elex 2797 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2797 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 p0ex 4196 . . . . . 6  |-  { (/) }  e.  _V
4 xpexg 4799 . . . . . 6  |-  ( ( A  e.  _V  /\  {
(/) }  e.  _V )  ->  ( A  X.  { (/) } )  e. 
_V )
53, 4mpan2 654 . . . . 5  |-  ( A  e.  _V  ->  ( A  X.  { (/) } )  e.  _V )
6 snex 4215 . . . . . 6  |-  { 1o }  e.  _V
7 xpexg 4799 . . . . . 6  |-  ( ( B  e.  _V  /\  { 1o }  e.  _V )  ->  ( B  X.  { 1o } )  e. 
_V )
86, 7mpan2 654 . . . . 5  |-  ( B  e.  _V  ->  ( B  X.  { 1o }
)  e.  _V )
95, 8anim12i 551 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( A  X.  { (/) } )  e. 
_V  /\  ( B  X.  { 1o } )  e.  _V ) )
10 unexb 4519 . . . 4  |-  ( ( ( A  X.  { (/)
} )  e.  _V  /\  ( B  X.  { 1o } )  e.  _V ) 
<->  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) )  e. 
_V )
119, 10sylib 190 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) )  e. 
_V )
12 xpeq1 4702 . . . . 5  |-  ( x  =  A  ->  (
x  X.  { (/) } )  =  ( A  X.  { (/) } ) )
1312uneq1d 3329 . . . 4  |-  ( x  =  A  ->  (
( x  X.  { (/)
} )  u.  (
y  X.  { 1o } ) )  =  ( ( A  X.  { (/) } )  u.  ( y  X.  { 1o } ) ) )
14 xpeq1 4702 . . . . 5  |-  ( y  =  B  ->  (
y  X.  { 1o } )  =  ( B  X.  { 1o } ) )
1514uneq2d 3330 . . . 4  |-  ( y  =  B  ->  (
( A  X.  { (/)
} )  u.  (
y  X.  { 1o } ) )  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
16 df-cda 7789 . . . 4  |-  +c  =  ( x  e.  _V ,  y  e.  _V  |->  ( ( x  X.  { (/) } )  u.  ( y  X.  { 1o } ) ) )
1713, 15, 16ovmpt2g 5943 . . 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 1280 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
191, 2, 18syl2an 465 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 6    /\ wa 360    = wceq 1624    e. wcel 1685   _Vcvv 2789    u. cun 3151   (/)c0 3456   {csn 3641    X. cxp 4686  (class class class)co 5819   1oc1o 6467    +c ccda 7788
This theorem is referenced by:  uncdadom  7792  cdaun  7793  cdaen  7794  cda1dif  7797  pm110.643  7798  xp2cda  7801  cdacomen  7802  cdaassen  7803  xpcdaen  7804  mapcdaen  7805  cdadom1  7807  cdaxpdom  7810  cdafi  7811  cdainf  7813  infcda1  7814  pwcdadom  7837  isfin4-3  7936  alephadd  8194  canthp1lem2  8270  xpsc  13453
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-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-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  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-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-cda 7789
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