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Theorem cdaen 7815
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 6884 . . . . . 6  |-  Rel  ~~
21brrelexi 4745 . . . . 5  |-  ( A 
~~  B  ->  A  e.  _V )
3 0ex 4166 . . . . 5  |-  (/)  e.  _V
4 xpsneng 6963 . . . . 5  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
52, 3, 4sylancl 643 . . . 4  |-  ( A 
~~  B  ->  ( A  X.  { (/) } ) 
~~  A )
61brrelex2i 4746 . . . . . . 7  |-  ( A 
~~  B  ->  B  e.  _V )
7 xpsneng 6963 . . . . . . 7  |-  ( ( B  e.  _V  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
86, 3, 7sylancl 643 . . . . . 6  |-  ( A 
~~  B  ->  ( B  X.  { (/) } ) 
~~  B )
9 ensym 6926 . . . . . 6  |-  ( ( B  X.  { (/) } )  ~~  B  ->  B  ~~  ( B  X.  { (/) } ) )
108, 9syl 15 . . . . 5  |-  ( A 
~~  B  ->  B  ~~  ( B  X.  { (/)
} ) )
11 entr 6929 . . . . 5  |-  ( ( A  ~~  B  /\  B  ~~  ( B  X.  { (/) } ) )  ->  A  ~~  ( B  X.  { (/) } ) )
1210, 11mpdan 649 . . . 4  |-  ( A 
~~  B  ->  A  ~~  ( B  X.  { (/)
} ) )
13 entr 6929 . . . 4  |-  ( ( ( A  X.  { (/)
} )  ~~  A  /\  A  ~~  ( B  X.  { (/) } ) )  ->  ( A  X.  { (/) } )  ~~  ( B  X.  { (/) } ) )
145, 12, 13syl2anc 642 . . 3  |-  ( A 
~~  B  ->  ( A  X.  { (/) } ) 
~~  ( B  X.  { (/) } ) )
151brrelexi 4745 . . . . 5  |-  ( C 
~~  D  ->  C  e.  _V )
16 1on 6502 . . . . 5  |-  1o  e.  On
17 xpsneng 6963 . . . . 5  |-  ( ( C  e.  _V  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
1815, 16, 17sylancl 643 . . . 4  |-  ( C 
~~  D  ->  ( C  X.  { 1o }
)  ~~  C )
191brrelex2i 4746 . . . . . . 7  |-  ( C 
~~  D  ->  D  e.  _V )
20 xpsneng 6963 . . . . . . 7  |-  ( ( D  e.  _V  /\  1o  e.  On )  -> 
( D  X.  { 1o } )  ~~  D
)
2119, 16, 20sylancl 643 . . . . . 6  |-  ( C 
~~  D  ->  ( D  X.  { 1o }
)  ~~  D )
22 ensym 6926 . . . . . 6  |-  ( ( D  X.  { 1o } )  ~~  D  ->  D  ~~  ( D  X.  { 1o }
) )
2321, 22syl 15 . . . . 5  |-  ( C 
~~  D  ->  D  ~~  ( D  X.  { 1o } ) )
24 entr 6929 . . . . 5  |-  ( ( C  ~~  D  /\  D  ~~  ( D  X.  { 1o } ) )  ->  C  ~~  ( D  X.  { 1o }
) )
2523, 24mpdan 649 . . . 4  |-  ( C 
~~  D  ->  C  ~~  ( D  X.  { 1o } ) )
26 entr 6929 . . . 4  |-  ( ( ( C  X.  { 1o } )  ~~  C  /\  C  ~~  ( D  X.  { 1o }
) )  ->  ( C  X.  { 1o }
)  ~~  ( D  X.  { 1o } ) )
2718, 25, 26syl2anc 642 . . 3  |-  ( C 
~~  D  ->  ( C  X.  { 1o }
)  ~~  ( D  X.  { 1o } ) )
28 xp01disj 6511 . . . 4  |-  ( ( A  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
29 xp01disj 6511 . . . 4  |-  ( ( B  X.  { (/) } )  i^i  ( D  X.  { 1o }
) )  =  (/)
30 unen 6959 . . . 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 }
) ) )
3128, 29, 30mpanr12 666 . . 3  |-  ( ( ( A  X.  { (/)
} )  ~~  ( B  X.  { (/) } )  /\  ( C  X.  { 1o } )  ~~  ( D  X.  { 1o } ) )  -> 
( ( A  X.  { (/) } )  u.  ( C  X.  { 1o } ) )  ~~  ( ( B  X.  { (/) } )  u.  ( D  X.  { 1o } ) ) )
3214, 27, 31syl2an 463 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( ( A  X.  { (/) } )  u.  ( C  X.  { 1o } ) )  ~~  ( ( B  X.  { (/) } )  u.  ( D  X.  { 1o } ) ) )
33 cdaval 7812 . . 3  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A  +c  C
)  =  ( ( A  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
342, 15, 33syl2an 463 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  +c  C
)  =  ( ( A  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
35 cdaval 7812 . . 3  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  ( B  +c  D
)  =  ( ( B  X.  { (/) } )  u.  ( D  X.  { 1o }
) ) )
366, 19, 35syl2an 463 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  +c  D
)  =  ( ( B  X.  { (/) } )  u.  ( D  X.  { 1o }
) ) )
3732, 34, 363brtr4d 4069 1  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  +c  C
)  ~~  ( B  +c  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   class class class wbr 4039   Oncon0 4408    X. cxp 4703  (class class class)co 5874   1oc1o 6488    ~~ cen 6876    +c ccda 7809
This theorem is referenced by:  cdaenun  7816  cardacda  7840  pwsdompw  7846  ackbij1lem5  7866  ackbij1lem9  7870  gchhar  8309
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-3or 935  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1o 6495  df-er 6676  df-en 6880  df-cda 7810
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