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Theorem cdaval 7812
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 8189, carddom 8192, and cardsdom 8193. 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 2809 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2809 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 p0ex 4213 . . . . . 6  |-  { (/) }  e.  _V
4 xpexg 4816 . . . . . 6  |-  ( ( A  e.  _V  /\  {
(/) }  e.  _V )  ->  ( A  X.  { (/) } )  e. 
_V )
53, 4mpan2 652 . . . . 5  |-  ( A  e.  _V  ->  ( A  X.  { (/) } )  e.  _V )
6 snex 4232 . . . . . 6  |-  { 1o }  e.  _V
7 xpexg 4816 . . . . . 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 4536 . . . 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 4719 . . . . 5  |-  ( x  =  A  ->  (
x  X.  { (/) } )  =  ( A  X.  { (/) } ) )
1312uneq1d 3341 . . . 4  |-  ( x  =  A  ->  (
( x  X.  { (/)
} )  u.  (
y  X.  { 1o } ) )  =  ( ( A  X.  { (/) } )  u.  ( y  X.  { 1o } ) ) )
14 xpeq1 4719 . . . . 5  |-  ( y  =  B  ->  (
y  X.  { 1o } )  =  ( B  X.  { 1o } ) )
1514uneq2d 3342 . . . 4  |-  ( y  =  B  ->  (
( A  X.  { (/)
} )  u.  (
y  X.  { 1o } ) )  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
16 df-cda 7810 . . . 4  |-  +c  =  ( x  e.  _V ,  y  e.  _V  |->  ( ( x  X.  { (/) } )  u.  ( y  X.  { 1o } ) ) )
1713, 15, 16ovmpt2g 5998 . . 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 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163   (/)c0 3468   {csn 3653    X. cxp 4703  (class class class)co 5874   1oc1o 6488    +c ccda 7809
This theorem is referenced by:  uncdadom  7813  cdaun  7814  cdaen  7815  cda1dif  7818  pm110.643  7819  xp2cda  7822  cdacomen  7823  cdaassen  7824  xpcdaen  7825  mapcdaen  7826  cdadom1  7828  cdaxpdom  7831  cdafi  7832  cdainf  7834  infcda1  7835  pwcdadom  7858  isfin4-3  7957  alephadd  8215  canthp1lem2  8291  xpsc  13475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-cda 7810
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