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Theorem xp2dju 7016
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 4554 . 2  |-  ( ( { (/) }  u.  { 1o } )  X.  A
)  =  ( ( { (/) }  X.  A
)  u.  ( { 1o }  X.  A
) )
2 df2o3 6279 . . . 4  |-  2o  =  { (/) ,  1o }
3 df-pr 3498 . . . 4  |-  { (/) ,  1o }  =  ( { (/) }  u.  { 1o } )
42, 3eqtri 2133 . . 3  |-  2o  =  ( { (/) }  u.  { 1o } )
54xpeq1i 4517 . 2  |-  ( 2o 
X.  A )  =  ( ( { (/) }  u.  { 1o }
)  X.  A )
6 df-dju 6873 . 2  |-  ( A A )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  A
) )
71, 5, 63eqtr4i 2143 1  |-  ( 2o 
X.  A )  =  ( A A )
Colors of variables: wff set class
Syntax hints:    = wceq 1312    u. cun 3033   (/)c0 3327   {csn 3491   {cpr 3492    X. cxp 4495   1oc1o 6258   2oc2o 6259   ⊔ cdju 6872
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-dif 3037  df-un 3039  df-nul 3328  df-pr 3498  df-opab 3948  df-suc 4251  df-xp 4503  df-1o 6265  df-2o 6266  df-dju 6873
This theorem is referenced by: (None)
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