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Theorem xp2dju 7244
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
xp2dju  |-  ( 2o 
X.  A )  =  ( A A )

Proof of Theorem xp2dju
StepHypRef Expression
1 xpundir 4701 . 2  |-  ( ( { (/) }  u.  { 1o } )  X.  A
)  =  ( ( { (/) }  X.  A
)  u.  ( { 1o }  X.  A
) )
2 df2o3 6455 . . . 4  |-  2o  =  { (/) ,  1o }
3 df-pr 3614 . . . 4  |-  { (/) ,  1o }  =  ( { (/) }  u.  { 1o } )
42, 3eqtri 2210 . . 3  |-  2o  =  ( { (/) }  u.  { 1o } )
54xpeq1i 4664 . 2  |-  ( 2o 
X.  A )  =  ( ( { (/) }  u.  { 1o }
)  X.  A )
6 df-dju 7067 . 2  |-  ( A A )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  A
) )
71, 5, 63eqtr4i 2220 1  |-  ( 2o 
X.  A )  =  ( A A )
Colors of variables: wff set class
Syntax hints:    = wceq 1364    u. cun 3142   (/)c0 3437   {csn 3607   {cpr 3608    X. cxp 4642   1oc1o 6434   2oc2o 6435   ⊔ cdju 7066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-un 3148  df-nul 3438  df-pr 3614  df-opab 4080  df-suc 4389  df-xp 4650  df-1o 6441  df-2o 6442  df-dju 7067
This theorem is referenced by: (None)
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