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

Theorem cdaen 8013
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 7077 . . . . . 6  |-  Rel  ~~
21brrelexi 4881 . . . . 5  |-  ( A 
~~  B  ->  A  e.  _V )
3 0ex 4303 . . . . 5  |-  (/)  e.  _V
4 xpsneng 7156 . . . . 5  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
52, 3, 4sylancl 644 . . . 4  |-  ( A 
~~  B  ->  ( A  X.  { (/) } ) 
~~  A )
61brrelex2i 4882 . . . . . . 7  |-  ( A 
~~  B  ->  B  e.  _V )
7 xpsneng 7156 . . . . . . 7  |-  ( ( B  e.  _V  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
86, 3, 7sylancl 644 . . . . . 6  |-  ( A 
~~  B  ->  ( B  X.  { (/) } ) 
~~  B )
98ensymd 7121 . . . . 5  |-  ( A 
~~  B  ->  B  ~~  ( B  X.  { (/)
} ) )
10 entr 7122 . . . . 5  |-  ( ( A  ~~  B  /\  B  ~~  ( B  X.  { (/) } ) )  ->  A  ~~  ( B  X.  { (/) } ) )
119, 10mpdan 650 . . . 4  |-  ( A 
~~  B  ->  A  ~~  ( B  X.  { (/)
} ) )
12 entr 7122 . . . 4  |-  ( ( ( A  X.  { (/)
} )  ~~  A  /\  A  ~~  ( B  X.  { (/) } ) )  ->  ( A  X.  { (/) } )  ~~  ( B  X.  { (/) } ) )
135, 11, 12syl2anc 643 . . 3  |-  ( A 
~~  B  ->  ( A  X.  { (/) } ) 
~~  ( B  X.  { (/) } ) )
141brrelexi 4881 . . . . 5  |-  ( C 
~~  D  ->  C  e.  _V )
15 1on 6694 . . . . 5  |-  1o  e.  On
16 xpsneng 7156 . . . . 5  |-  ( ( C  e.  _V  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
1714, 15, 16sylancl 644 . . . 4  |-  ( C 
~~  D  ->  ( C  X.  { 1o }
)  ~~  C )
181brrelex2i 4882 . . . . . . 7  |-  ( C 
~~  D  ->  D  e.  _V )
19 xpsneng 7156 . . . . . . 7  |-  ( ( D  e.  _V  /\  1o  e.  On )  -> 
( D  X.  { 1o } )  ~~  D
)
2018, 15, 19sylancl 644 . . . . . 6  |-  ( C 
~~  D  ->  ( D  X.  { 1o }
)  ~~  D )
2120ensymd 7121 . . . . 5  |-  ( C 
~~  D  ->  D  ~~  ( D  X.  { 1o } ) )
22 entr 7122 . . . . 5  |-  ( ( C  ~~  D  /\  D  ~~  ( D  X.  { 1o } ) )  ->  C  ~~  ( D  X.  { 1o }
) )
2321, 22mpdan 650 . . . 4  |-  ( C 
~~  D  ->  C  ~~  ( D  X.  { 1o } ) )
24 entr 7122 . . . 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 6703 . . . 4  |-  ( ( A  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
27 xp01disj 6703 . . . 4  |-  ( ( B  X.  { (/) } )  i^i  ( D  X.  { 1o }
) )  =  (/)
28 unen 7152 . . . 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 8010 . . 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 8010 . . 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 4206 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 2920    u. cun 3282    i^i cin 3283   (/)c0 3592   {csn 3778   class class class wbr 4176   Oncon0 4545    X. cxp 4839  (class class class)co 6044   1oc1o 6680    ~~ cen 7069    +c ccda 8007
This theorem is referenced by:  cdaenun  8014  cardacda  8038  pwsdompw  8044  ackbij1lem5  8064  ackbij1lem9  8068  gchhar  8506
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 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-suc 4551  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1o 6687  df-er 6868  df-en 7073  df-cda 8008
  Copyright terms: Public domain W3C validator