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Theorem xp2cda 8020
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 8010 . . 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 6700 . . . . 5  |-  2o  =  { (/) ,  1o }
4 df-pr 3785 . . . . 5  |-  { (/) ,  1o }  =  ( { (/) }  u.  { 1o } )
53, 4eqtri 2428 . . . 4  |-  2o  =  ( { (/) }  u.  { 1o } )
65xpeq2i 4862 . . 3  |-  ( A  X.  2o )  =  ( A  X.  ( { (/) }  u.  { 1o } ) )
7 xpundi 4893 . . 3  |-  ( A  X.  ( { (/) }  u.  { 1o }
) )  =  ( ( A  X.  { (/)
} )  u.  ( A  X.  { 1o }
) )
86, 7eqtri 2428 . 2  |-  ( A  X.  2o )  =  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o } ) )
92, 8syl6reqr 2459 1  |-  ( A  e.  V  ->  ( A  X.  2o )  =  ( A  +c  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    u. cun 3282   (/)c0 3592   {csn 3778   {cpr 3779    X. cxp 4839  (class class class)co 6044   1oc1o 6680   2oc2o 6681    +c ccda 8007
This theorem is referenced by:  pwcda1  8034  unctb  8045  infcdaabs  8046  ackbij1lem5  8064  fin56  8233
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-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-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-suc 4551  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1o 6687  df-2o 6688  df-cda 8008
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