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Theorem xp2cda 7993
Description: Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
xp2cda  |-  ( A  e.  V  ->  ( A  X.  2o )  =  ( A  +c  A
) )

Proof of Theorem xp2cda
StepHypRef Expression
1 cdaval 7983 . . 3  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  +c  A
)  =  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o }
) ) )
21anidms 627 . 2  |-  ( A  e.  V  ->  ( A  +c  A )  =  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o } ) ) )
3 df2o3 6673 . . . . 5  |-  2o  =  { (/) ,  1o }
4 df-pr 3764 . . . . 5  |-  { (/) ,  1o }  =  ( { (/) }  u.  { 1o } )
53, 4eqtri 2407 . . . 4  |-  2o  =  ( { (/) }  u.  { 1o } )
65xpeq2i 4839 . . 3  |-  ( A  X.  2o )  =  ( A  X.  ( { (/) }  u.  { 1o } ) )
7 xpundi 4870 . . 3  |-  ( A  X.  ( { (/) }  u.  { 1o }
) )  =  ( ( A  X.  { (/)
} )  u.  ( A  X.  { 1o }
) )
86, 7eqtri 2407 . 2  |-  ( A  X.  2o )  =  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o } ) )
92, 8syl6reqr 2438 1  |-  ( A  e.  V  ->  ( A  X.  2o )  =  ( A  +c  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    u. cun 3261   (/)c0 3571   {csn 3757   {cpr 3758    X. cxp 4816  (class class class)co 6020   1oc1o 6653   2oc2o 6654    +c ccda 7980
This theorem is referenced by:  pwcda1  8007  unctb  8018  infcdaabs  8019  ackbij1lem5  8037  fin56  8206
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-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-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-suc 4528  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1o 6660  df-2o 6661  df-cda 7981
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