MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cdaen Unicode version

Theorem cdaen 8009
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 7073 . . . . . 6  |-  Rel  ~~
21brrelexi 4877 . . . . 5  |-  ( A 
~~  B  ->  A  e.  _V )
3 0ex 4299 . . . . 5  |-  (/)  e.  _V
4 xpsneng 7152 . . . . 5  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
52, 3, 4sylancl 644 . . . 4  |-  ( A 
~~  B  ->  ( A  X.  { (/) } ) 
~~  A )
61brrelex2i 4878 . . . . . . 7  |-  ( A 
~~  B  ->  B  e.  _V )
7 xpsneng 7152 . . . . . . 7  |-  ( ( B  e.  _V  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
86, 3, 7sylancl 644 . . . . . 6  |-  ( A 
~~  B  ->  ( B  X.  { (/) } ) 
~~  B )
98ensymd 7117 . . . . 5  |-  ( A 
~~  B  ->  B  ~~  ( B  X.  { (/)
} ) )
10 entr 7118 . . . . 5  |-  ( ( A  ~~  B  /\  B  ~~  ( B  X.  { (/) } ) )  ->  A  ~~  ( B  X.  { (/) } ) )
119, 10mpdan 650 . . . 4  |-  ( A 
~~  B  ->  A  ~~  ( B  X.  { (/)
} ) )
12 entr 7118 . . . 4  |-  ( ( ( A  X.  { (/)
} )  ~~  A  /\  A  ~~  ( B  X.  { (/) } ) )  ->  ( A  X.  { (/) } )  ~~  ( B  X.  { (/) } ) )
135, 11, 12syl2anc 643 . . 3  |-  ( A 
~~  B  ->  ( A  X.  { (/) } ) 
~~  ( B  X.  { (/) } ) )
141brrelexi 4877 . . . . 5  |-  ( C 
~~  D  ->  C  e.  _V )
15 1on 6690 . . . . 5  |-  1o  e.  On
16 xpsneng 7152 . . . . 5  |-  ( ( C  e.  _V  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
1714, 15, 16sylancl 644 . . . 4  |-  ( C 
~~  D  ->  ( C  X.  { 1o }
)  ~~  C )
181brrelex2i 4878 . . . . . . 7  |-  ( C 
~~  D  ->  D  e.  _V )
19 xpsneng 7152 . . . . . . 7  |-  ( ( D  e.  _V  /\  1o  e.  On )  -> 
( D  X.  { 1o } )  ~~  D
)
2018, 15, 19sylancl 644 . . . . . 6  |-  ( C 
~~  D  ->  ( D  X.  { 1o }
)  ~~  D )
2120ensymd 7117 . . . . 5  |-  ( C 
~~  D  ->  D  ~~  ( D  X.  { 1o } ) )
22 entr 7118 . . . . 5  |-  ( ( C  ~~  D  /\  D  ~~  ( D  X.  { 1o } ) )  ->  C  ~~  ( D  X.  { 1o }
) )
2321, 22mpdan 650 . . . 4  |-  ( C 
~~  D  ->  C  ~~  ( D  X.  { 1o } ) )
24 entr 7118 . . . 4  |-  ( ( ( C  X.  { 1o } )  ~~  C  /\  C  ~~  ( D  X.  { 1o }
) )  ->  ( C  X.  { 1o }
)  ~~  ( D  X.  { 1o } ) )
2517, 23, 24syl2anc 643 . . 3  |-  ( C 
~~  D  ->  ( C  X.  { 1o }
)  ~~  ( D  X.  { 1o } ) )
26 xp01disj 6699 . . . 4  |-  ( ( A  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
27 xp01disj 6699 . . . 4  |-  ( ( B  X.  { (/) } )  i^i  ( D  X.  { 1o }
) )  =  (/)
28 unen 7148 . . . 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 }
) ) )
2926, 27, 28mpanr12 667 . . 3  |-  ( ( ( A  X.  { (/)
} )  ~~  ( B  X.  { (/) } )  /\  ( C  X.  { 1o } )  ~~  ( D  X.  { 1o } ) )  -> 
( ( A  X.  { (/) } )  u.  ( C  X.  { 1o } ) )  ~~  ( ( B  X.  { (/) } )  u.  ( D  X.  { 1o } ) ) )
3013, 25, 29syl2an 464 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( ( A  X.  { (/) } )  u.  ( C  X.  { 1o } ) )  ~~  ( ( B  X.  { (/) } )  u.  ( D  X.  { 1o } ) ) )
31 cdaval 8006 . . 3  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A  +c  C
)  =  ( ( A  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
322, 14, 31syl2an 464 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  +c  C
)  =  ( ( A  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
33 cdaval 8006 . . 3  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  ( B  +c  D
)  =  ( ( B  X.  { (/) } )  u.  ( D  X.  { 1o }
) ) )
346, 18, 33syl2an 464 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  +c  D
)  =  ( ( B  X.  { (/) } )  u.  ( D  X.  { 1o }
) ) )
3530, 32, 343brtr4d 4202 1  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  +c  C
)  ~~  ( B  +c  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278    i^i cin 3279   (/)c0 3588   {csn 3774   class class class wbr 4172   Oncon0 4541    X. cxp 4835  (class class class)co 6040   1oc1o 6676    ~~ cen 7065    +c ccda 8003
This theorem is referenced by:  cdaenun  8010  cardacda  8034  pwsdompw  8040  ackbij1lem5  8060  ackbij1lem9  8064  gchhar  8502
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1o 6683  df-er 6864  df-en 7069  df-cda 8004
  Copyright terms: Public domain W3C validator