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Theorem cdaun 7814
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 7812 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
213adant3 975 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  ( A  +c  B )  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
3 0ex 4166 . . . . . 6  |-  (/)  e.  _V
4 xpsneng 6963 . . . . . 6  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
53, 4mpan2 652 . . . . 5  |-  ( A  e.  V  ->  ( A  X.  { (/) } ) 
~~  A )
6 1on 6502 . . . . . 6  |-  1o  e.  On
7 xpsneng 6963 . . . . . 6  |-  ( ( B  e.  W  /\  1o  e.  On )  -> 
( B  X.  { 1o } )  ~~  B
)
86, 7mpan2 652 . . . . 5  |-  ( B  e.  W  ->  ( B  X.  { 1o }
)  ~~  B )
95, 8anim12i 549 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  X.  { (/) } )  ~~  A  /\  ( B  X.  { 1o } )  ~~  B ) )
10 xp01disj 6511 . . . . 5  |-  ( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o }
) )  =  (/)
1110jctl 525 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( ( A  X.  { (/)
} )  i^i  ( B  X.  { 1o }
) )  =  (/)  /\  ( A  i^i  B
)  =  (/) ) )
12 unen 6959 . . . 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 463 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( A  i^i  B )  =  (/) )  ->  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) 
~~  ( A  u.  B ) )
14133impa 1146 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  (
( A  X.  { (/)
} )  u.  ( B  X.  { 1o }
) )  ~~  ( A  u.  B )
)
152, 14eqbrtrd 4059 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 358    /\ w3a 934    = 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  cda0en  7821  ficardun  7844  ackbij1lem9  7870  canthp1lem1  8290
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-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-en 6880  df-cda 7810
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