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Theorem cdaen 7986
Description: Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
cdaen  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  +c  C
)  ~~  ( B  +c  D ) )

Proof of Theorem cdaen
StepHypRef Expression
1 relen 7050 . . . . . 6  |-  Rel  ~~
21brrelexi 4858 . . . . 5  |-  ( A 
~~  B  ->  A  e.  _V )
3 0ex 4280 . . . . 5  |-  (/)  e.  _V
4 xpsneng 7129 . . . . 5  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
52, 3, 4sylancl 644 . . . 4  |-  ( A 
~~  B  ->  ( A  X.  { (/) } ) 
~~  A )
61brrelex2i 4859 . . . . . . 7  |-  ( A 
~~  B  ->  B  e.  _V )
7 xpsneng 7129 . . . . . . 7  |-  ( ( B  e.  _V  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
86, 3, 7sylancl 644 . . . . . 6  |-  ( A 
~~  B  ->  ( B  X.  { (/) } ) 
~~  B )
98ensymd 7094 . . . . 5  |-  ( A 
~~  B  ->  B  ~~  ( B  X.  { (/)
} ) )
10 entr 7095 . . . . 5  |-  ( ( A  ~~  B  /\  B  ~~  ( B  X.  { (/) } ) )  ->  A  ~~  ( B  X.  { (/) } ) )
119, 10mpdan 650 . . . 4  |-  ( A 
~~  B  ->  A  ~~  ( B  X.  { (/)
} ) )
12 entr 7095 . . . 4  |-  ( ( ( A  X.  { (/)
} )  ~~  A  /\  A  ~~  ( B  X.  { (/) } ) )  ->  ( A  X.  { (/) } )  ~~  ( B  X.  { (/) } ) )
135, 11, 12syl2anc 643 . . 3  |-  ( A 
~~  B  ->  ( A  X.  { (/) } ) 
~~  ( B  X.  { (/) } ) )
141brrelexi 4858 . . . . 5  |-  ( C 
~~  D  ->  C  e.  _V )
15 1on 6667 . . . . 5  |-  1o  e.  On
16 xpsneng 7129 . . . . 5  |-  ( ( C  e.  _V  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
1714, 15, 16sylancl 644 . . . 4  |-  ( C 
~~  D  ->  ( C  X.  { 1o }
)  ~~  C )
181brrelex2i 4859 . . . . . . 7  |-  ( C 
~~  D  ->  D  e.  _V )
19 xpsneng 7129 . . . . . . 7  |-  ( ( D  e.  _V  /\  1o  e.  On )  -> 
( D  X.  { 1o } )  ~~  D
)
2018, 15, 19sylancl 644 . . . . . 6  |-  ( C 
~~  D  ->  ( D  X.  { 1o }
)  ~~  D )
2120ensymd 7094 . . . . 5  |-  ( C 
~~  D  ->  D  ~~  ( D  X.  { 1o } ) )
22 entr 7095 . . . . 5  |-  ( ( C  ~~  D  /\  D  ~~  ( D  X.  { 1o } ) )  ->  C  ~~  ( D  X.  { 1o }
) )
2321, 22mpdan 650 . . . 4  |-  ( C 
~~  D  ->  C  ~~  ( D  X.  { 1o } ) )
24 entr 7095 . . . 4  |-  ( ( ( C  X.  { 1o } )  ~~  C  /\  C  ~~  ( D  X.  { 1o }
) )  ->  ( C  X.  { 1o }
)  ~~  ( D  X.  { 1o } ) )
2517, 23, 24syl2anc 643 . . 3  |-  ( C 
~~  D  ->  ( C  X.  { 1o }
)  ~~  ( D  X.  { 1o } ) )
26 xp01disj 6676 . . . 4  |-  ( ( A  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
27 xp01disj 6676 . . . 4  |-  ( ( B  X.  { (/) } )  i^i  ( D  X.  { 1o }
) )  =  (/)
28 unen 7125 . . . 4  |-  ( ( ( ( A  X.  { (/) } )  ~~  ( B  X.  { (/) } )  /\  ( C  X.  { 1o }
)  ~~  ( D  X.  { 1o } ) )  /\  ( ( ( A  X.  { (/)
} )  i^i  ( C  X.  { 1o }
) )  =  (/)  /\  ( ( B  X.  { (/) } )  i^i  ( D  X.  { 1o } ) )  =  (/) ) )  ->  (
( A  X.  { (/)
} )  u.  ( C  X.  { 1o }
) )  ~~  (
( B  X.  { (/)
} )  u.  ( D  X.  { 1o }
) ) )
2926, 27, 28mpanr12 667 . . 3  |-  ( ( ( A  X.  { (/)
} )  ~~  ( B  X.  { (/) } )  /\  ( C  X.  { 1o } )  ~~  ( D  X.  { 1o } ) )  -> 
( ( A  X.  { (/) } )  u.  ( C  X.  { 1o } ) )  ~~  ( ( B  X.  { (/) } )  u.  ( D  X.  { 1o } ) ) )
3013, 25, 29syl2an 464 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( ( A  X.  { (/) } )  u.  ( C  X.  { 1o } ) )  ~~  ( ( B  X.  { (/) } )  u.  ( D  X.  { 1o } ) ) )
31 cdaval 7983 . . 3  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A  +c  C
)  =  ( ( A  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
322, 14, 31syl2an 464 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  +c  C
)  =  ( ( A  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
33 cdaval 7983 . . 3  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  ( B  +c  D
)  =  ( ( B  X.  { (/) } )  u.  ( D  X.  { 1o }
) ) )
346, 18, 33syl2an 464 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  +c  D
)  =  ( ( B  X.  { (/) } )  u.  ( D  X.  { 1o }
) ) )
3530, 32, 343brtr4d 4183 1  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  +c  C
)  ~~  ( B  +c  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899    u. cun 3261    i^i cin 3262   (/)c0 3571   {csn 3757   class class class wbr 4153   Oncon0 4522    X. cxp 4816  (class class class)co 6020   1oc1o 6653    ~~ cen 7042    +c ccda 7980
This theorem is referenced by:  cdaenun  7987  cardacda  8011  pwsdompw  8017  ackbij1lem5  8037  ackbij1lem9  8041  gchhar  8479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-suc 4528  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1o 6660  df-er 6841  df-en 7046  df-cda 7981
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