Theorem xp2dju 7277

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

Step	Hyp	Ref	Expression
1		xpundir 4717	. 2 |- ( ( { (/) } u. { 1o } ) X. A ) = ( ( { (/) } X. A ) u. ( { 1o } X. A ) )
2		df2o3 6485	. . . 4 |- 2o = { (/) , 1o }
3		df-pr 3626	. . . 4 |- { (/) , 1o } = ( { (/) } u. { 1o } )
4	2, 3	eqtri 2214	. . 3 |- 2o = ( { (/) } u. { 1o } )
5	4	xpeq1i 4680	. 2 |- ( 2o X. A ) = ( ( { (/) } u. { 1o } ) X. A )
6		df-dju 7099	. 2 |- ( A ⊔ A ) = ( ( { (/) } X. A ) u. ( { 1o } X. A ) )
7	1, 5, 6	3eqtr4i 2224	1 |- ( 2o X. A ) = ( A ⊔ A )

Syntax hints:   = wceq 1364   u. cun 3152   (/)c0 3447   {csn 3619   {cpr 3620   X. cxp 4658   1oc1o 6464   2oc2o 6465   ⊔ cdju 7098

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 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3156  df-un 3158  df-nul 3448  df-pr 3626  df-opab 4092  df-suc 4403  df-xp 4666  df-1o 6471  df-2o 6472  df-dju 7099

This theorem is referenced by: (None)