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Theorem cdaenun 7800
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 7799 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  +c  C
)  ~~  ( B  +c  D ) )
213adant3 975 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D  /\  ( B  i^i  D )  =  (/) )  ->  ( A  +c  C )  ~~  ( B  +c  D
) )
3 relen 6868 . . . 4  |-  Rel  ~~
43brrelex2i 4730 . . 3  |-  ( A 
~~  B  ->  B  e.  _V )
53brrelex2i 4730 . . 3  |-  ( C 
~~  D  ->  D  e.  _V )
6 id 19 . . 3  |-  ( ( B  i^i  D )  =  (/)  ->  ( B  i^i  D )  =  (/) )
7 cdaun 7798 . . 3  |-  ( ( B  e.  _V  /\  D  e.  _V  /\  ( B  i^i  D )  =  (/) )  ->  ( B  +c  D )  ~~  ( B  u.  D
) )
84, 5, 6, 7syl3an 1224 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D  /\  ( B  i^i  D )  =  (/) )  ->  ( B  +c  D )  ~~  ( B  u.  D
) )
9 entr 6913 . 2  |-  ( ( ( A  +c  C
)  ~~  ( B  +c  D )  /\  ( B  +c  D )  ~~  ( B  u.  D
) )  ->  ( A  +c  C )  ~~  ( B  u.  D
) )
102, 8, 9syl2anc 642 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 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150    i^i cin 3151   (/)c0 3455   class class class wbr 4023  (class class class)co 5858    ~~ cen 6860    +c ccda 7793
This theorem is referenced by:  cda1en  7801  cdacomen  7807  cdaassen  7808  xpcdaen  7809  onacda  7823  pwxpndom2  8287
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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