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Theorem xp2cda 8052
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 8042 . . 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 6729 . . . . 5  |-  2o  =  { (/) ,  1o }
4 df-pr 3813 . . . . 5  |-  { (/) ,  1o }  =  ( { (/) }  u.  { 1o } )
53, 4eqtri 2455 . . . 4  |-  2o  =  ( { (/) }  u.  { 1o } )
65xpeq2i 4891 . . 3  |-  ( A  X.  2o )  =  ( A  X.  ( { (/) }  u.  { 1o } ) )
7 xpundi 4922 . . 3  |-  ( A  X.  ( { (/) }  u.  { 1o }
) )  =  ( ( A  X.  { (/)
} )  u.  ( A  X.  { 1o }
) )
86, 7eqtri 2455 . 2  |-  ( A  X.  2o )  =  ( ( A  X.  { (/) } )  u.  ( A  X.  { 1o } ) )
92, 8syl6reqr 2486 1  |-  ( A  e.  V  ->  ( A  X.  2o )  =  ( A  +c  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    u. cun 3310   (/)c0 3620   {csn 3806   {cpr 3807    X. cxp 4868  (class class class)co 6073   1oc1o 6709   2oc2o 6710    +c ccda 8039
This theorem is referenced by:  pwcda1  8066  unctb  8077  infcdaabs  8078  ackbij1lem5  8096  fin56  8265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1o 6716  df-2o 6717  df-cda 8040
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