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Theorem cdaen 7799
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 6868 . . . . . 6  |-  Rel  ~~
21brrelexi 4729 . . . . 5  |-  ( A 
~~  B  ->  A  e.  _V )
3 0ex 4150 . . . . 5  |-  (/)  e.  _V
4 xpsneng 6947 . . . . 5  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
52, 3, 4sylancl 643 . . . 4  |-  ( A 
~~  B  ->  ( A  X.  { (/) } ) 
~~  A )
61brrelex2i 4730 . . . . . . 7  |-  ( A 
~~  B  ->  B  e.  _V )
7 xpsneng 6947 . . . . . . 7  |-  ( ( B  e.  _V  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
86, 3, 7sylancl 643 . . . . . 6  |-  ( A 
~~  B  ->  ( B  X.  { (/) } ) 
~~  B )
9 ensym 6910 . . . . . 6  |-  ( ( B  X.  { (/) } )  ~~  B  ->  B  ~~  ( B  X.  { (/) } ) )
108, 9syl 15 . . . . 5  |-  ( A 
~~  B  ->  B  ~~  ( B  X.  { (/)
} ) )
11 entr 6913 . . . . 5  |-  ( ( A  ~~  B  /\  B  ~~  ( B  X.  { (/) } ) )  ->  A  ~~  ( B  X.  { (/) } ) )
1210, 11mpdan 649 . . . 4  |-  ( A 
~~  B  ->  A  ~~  ( B  X.  { (/)
} ) )
13 entr 6913 . . . 4  |-  ( ( ( A  X.  { (/)
} )  ~~  A  /\  A  ~~  ( B  X.  { (/) } ) )  ->  ( A  X.  { (/) } )  ~~  ( B  X.  { (/) } ) )
145, 12, 13syl2anc 642 . . 3  |-  ( A 
~~  B  ->  ( A  X.  { (/) } ) 
~~  ( B  X.  { (/) } ) )
151brrelexi 4729 . . . . 5  |-  ( C 
~~  D  ->  C  e.  _V )
16 1on 6486 . . . . 5  |-  1o  e.  On
17 xpsneng 6947 . . . . 5  |-  ( ( C  e.  _V  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
1815, 16, 17sylancl 643 . . . 4  |-  ( C 
~~  D  ->  ( C  X.  { 1o }
)  ~~  C )
191brrelex2i 4730 . . . . . . 7  |-  ( C 
~~  D  ->  D  e.  _V )
20 xpsneng 6947 . . . . . . 7  |-  ( ( D  e.  _V  /\  1o  e.  On )  -> 
( D  X.  { 1o } )  ~~  D
)
2119, 16, 20sylancl 643 . . . . . 6  |-  ( C 
~~  D  ->  ( D  X.  { 1o }
)  ~~  D )
22 ensym 6910 . . . . . 6  |-  ( ( D  X.  { 1o } )  ~~  D  ->  D  ~~  ( D  X.  { 1o }
) )
2321, 22syl 15 . . . . 5  |-  ( C 
~~  D  ->  D  ~~  ( D  X.  { 1o } ) )
24 entr 6913 . . . . 5  |-  ( ( C  ~~  D  /\  D  ~~  ( D  X.  { 1o } ) )  ->  C  ~~  ( D  X.  { 1o }
) )
2523, 24mpdan 649 . . . 4  |-  ( C 
~~  D  ->  C  ~~  ( D  X.  { 1o } ) )
26 entr 6913 . . . 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 6495 . . . 4  |-  ( ( A  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
29 xp01disj 6495 . . . 4  |-  ( ( B  X.  { (/) } )  i^i  ( D  X.  { 1o }
) )  =  (/)
30 unen 6943 . . . 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 7796 . . 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 7796 . . 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 4053 1  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  +c  C
)  ~~  ( B  +c  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640   class class class wbr 4023   Oncon0 4392    X. cxp 4687  (class class class)co 5858   1oc1o 6472    ~~ cen 6860    +c ccda 7793
This theorem is referenced by:  cdaenun  7800  cardacda  7824  pwsdompw  7830  ackbij1lem5  7850  ackbij1lem9  7854  gchhar  8293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1o 6479  df-er 6660  df-en 6864  df-cda 7794
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