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Theorem cdaenun 7988
Description: Cardinal addition is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
cdaenun  |-  ( ( A  ~~  B  /\  C  ~~  D  /\  ( B  i^i  D )  =  (/) )  ->  ( A  +c  C )  ~~  ( B  u.  D
) )

Proof of Theorem cdaenun
StepHypRef Expression
1 cdaen 7987 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  +c  C
)  ~~  ( B  +c  D ) )
213adant3 977 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D  /\  ( B  i^i  D )  =  (/) )  ->  ( A  +c  C )  ~~  ( B  +c  D
) )
3 relen 7051 . . . 4  |-  Rel  ~~
43brrelex2i 4860 . . 3  |-  ( A 
~~  B  ->  B  e.  _V )
53brrelex2i 4860 . . 3  |-  ( C 
~~  D  ->  D  e.  _V )
6 id 20 . . 3  |-  ( ( B  i^i  D )  =  (/)  ->  ( B  i^i  D )  =  (/) )
7 cdaun 7986 . . 3  |-  ( ( B  e.  _V  /\  D  e.  _V  /\  ( B  i^i  D )  =  (/) )  ->  ( B  +c  D )  ~~  ( B  u.  D
) )
84, 5, 6, 7syl3an 1226 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D  /\  ( B  i^i  D )  =  (/) )  ->  ( B  +c  D )  ~~  ( B  u.  D
) )
9 entr 7096 . 2  |-  ( ( ( A  +c  C
)  ~~  ( B  +c  D )  /\  ( B  +c  D )  ~~  ( B  u.  D
) )  ->  ( A  +c  C )  ~~  ( B  u.  D
) )
102, 8, 9syl2anc 643 1  |-  ( ( A  ~~  B  /\  C  ~~  D  /\  ( B  i^i  D )  =  (/) )  ->  ( A  +c  C )  ~~  ( B  u.  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2900    u. cun 3262    i^i cin 3263   (/)c0 3572   class class class wbr 4154  (class class class)co 6021    ~~ cen 7043    +c ccda 7981
This theorem is referenced by:  cda1en  7989  cdacomen  7995  cdaassen  7996  xpcdaen  7997  onacda  8011  pwxpndom2  8474
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-suc 4529  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1o 6661  df-er 6842  df-en 7047  df-cda 7982
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