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Theorem xp2cda 7808
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 7798 . . 3  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  +c  A
)  =  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o }
) ) )
21anidms 626 . 2  |-  ( A  e.  V  ->  ( A  +c  A )  =  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o } ) ) )
3 df2o3 6494 . . . . 5  |-  2o  =  { (/) ,  1o }
4 df-pr 3649 . . . . 5  |-  { (/) ,  1o }  =  ( { (/) }  u.  { 1o } )
53, 4eqtri 2305 . . . 4  |-  2o  =  ( { (/) }  u.  { 1o } )
65xpeq2i 4712 . . 3  |-  ( A  X.  2o )  =  ( A  X.  ( { (/) }  u.  { 1o } ) )
7 xpundi 4743 . . 3  |-  ( A  X.  ( { (/) }  u.  { 1o }
) )  =  ( ( A  X.  { (/)
} )  u.  ( A  X.  { 1o }
) )
86, 7eqtri 2305 . 2  |-  ( A  X.  2o )  =  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o } ) )
92, 8syl6reqr 2336 1  |-  ( A  e.  V  ->  ( A  X.  2o )  =  ( A  +c  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1625    e. wcel 1686    u. cun 3152   (/)c0 3457   {csn 3642   {cpr 3643    X. cxp 4689  (class class class)co 5860   1oc1o 6474   2oc2o 6475    +c ccda 7795
This theorem is referenced by:  pwcda1  7822  unctb  7833  infcdaabs  7834  ackbij1lem5  7852  fin56  8021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4311  df-suc 4400  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-iota 5221  df-fun 5259  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1o 6481  df-2o 6482  df-cda 7796
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