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Theorem cdaval 7680
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 8055, carddom 8058, and cardsdom 8059. 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
StepHypRef Expression
1 elex 2735 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2735 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 p0ex 4091 . . . . . 6  |-  { (/) }  e.  _V
4 xpexg 4707 . . . . . 6  |-  ( ( A  e.  _V  /\  {
(/) }  e.  _V )  ->  ( A  X.  { (/) } )  e. 
_V )
53, 4mpan2 655 . . . . 5  |-  ( A  e.  _V  ->  ( A  X.  { (/) } )  e.  _V )
6 snex 4110 . . . . . 6  |-  { 1o }  e.  _V
7 xpexg 4707 . . . . . 6  |-  ( ( B  e.  _V  /\  { 1o }  e.  _V )  ->  ( B  X.  { 1o } )  e. 
_V )
86, 7mpan2 655 . . . . 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 4411 . . . 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 4610 . . . . 5  |-  ( x  =  A  ->  (
x  X.  { (/) } )  =  ( A  X.  { (/) } ) )
1312uneq1d 3238 . . . 4  |-  ( x  =  A  ->  (
( x  X.  { (/)
} )  u.  (
y  X.  { 1o } ) )  =  ( ( A  X.  { (/) } )  u.  ( y  X.  { 1o } ) ) )
14 xpeq1 4610 . . . . 5  |-  ( y  =  B  ->  (
y  X.  { 1o } )  =  ( B  X.  { 1o } ) )
1514uneq2d 3239 . . . 4  |-  ( y  =  B  ->  (
( A  X.  { (/)
} )  u.  (
y  X.  { 1o } ) )  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
16 df-cda 7678 . . . 4  |-  +c  =  ( x  e.  _V ,  y  e.  _V  |->  ( ( x  X.  { (/) } )  u.  ( y  X.  { 1o } ) ) )
1713, 15, 16ovmpt2g 5834 . . 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 1283 . 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 1619    e. wcel 1621   _Vcvv 2727    u. cun 3076   (/)c0 3362   {csn 3544    X. cxp 4578  (class class class)co 5710   1oc1o 6358    +c ccda 7677
This theorem is referenced by:  uncdadom  7681  cdaun  7682  cdaen  7683  cda1dif  7686  pm110.643  7687  xp2cda  7690  cdacomen  7691  cdaassen  7692  xpcdaen  7693  mapcdaen  7694  cdadom1  7696  cdaxpdom  7699  cdafi  7700  cdainf  7702  infcda1  7703  pwcdadom  7726  isfin4-3  7825  alephadd  8079  canthp1lem2  8155  xpsc  13333
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-cda 7678
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