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Theorem cdaun 7985
Description: Cardinal addition is equinumerous to union for disjoint sets. (Contributed by NM, 5-Apr-2007.)
Assertion
Ref Expression
cdaun  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  ( A  +c  B )  ~~  ( A  u.  B
) )

Proof of Theorem cdaun
StepHypRef Expression
1 cdaval 7983 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
213adant3 977 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  ( A  +c  B )  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
3 0ex 4280 . . . . . 6  |-  (/)  e.  _V
4 xpsneng 7129 . . . . . 6  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
53, 4mpan2 653 . . . . 5  |-  ( A  e.  V  ->  ( A  X.  { (/) } ) 
~~  A )
6 1on 6667 . . . . . 6  |-  1o  e.  On
7 xpsneng 7129 . . . . . 6  |-  ( ( B  e.  W  /\  1o  e.  On )  -> 
( B  X.  { 1o } )  ~~  B
)
86, 7mpan2 653 . . . . 5  |-  ( B  e.  W  ->  ( B  X.  { 1o }
)  ~~  B )
95, 8anim12i 550 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  X.  { (/) } )  ~~  A  /\  ( B  X.  { 1o } )  ~~  B ) )
10 xp01disj 6676 . . . . 5  |-  ( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o }
) )  =  (/)
1110jctl 526 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( ( A  X.  { (/)
} )  i^i  ( B  X.  { 1o }
) )  =  (/)  /\  ( A  i^i  B
)  =  (/) ) )
12 unen 7125 . . . 4  |-  ( ( ( ( A  X.  { (/) } )  ~~  A  /\  ( B  X.  { 1o } )  ~~  B )  /\  (
( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o } ) )  =  (/)  /\  ( A  i^i  B )  =  (/) ) )  ->  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) 
~~  ( A  u.  B ) )
139, 11, 12syl2an 464 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( A  i^i  B )  =  (/) )  ->  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) 
~~  ( A  u.  B ) )
14133impa 1148 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  (
( A  X.  { (/)
} )  u.  ( B  X.  { 1o }
) )  ~~  ( A  u.  B )
)
152, 14eqbrtrd 4173 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  ( A  +c  B )  ~~  ( A  u.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = 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  cda0en  7992  ficardun  8015  ackbij1lem9  8041  canthp1lem1  8460
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-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-en 7046  df-cda 7981
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