Theorem xp2dju 7393

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 4775	. 2 |- ( ( { (/) } u. { 1o } ) X. A ) = ( ( { (/) } X. A ) u. ( { 1o } X. A ) )
2		df2o3 6574	. . . 4 |- 2o = { (/) , 1o }
3		df-pr 3673	. . . 4 |- { (/) , 1o } = ( { (/) } u. { 1o } )
4	2, 3	eqtri 2250	. . 3 |- 2o = ( { (/) } u. { 1o } )
5	4	xpeq1i 4738	. 2 |- ( 2o X. A ) = ( ( { (/) } u. { 1o } ) X. A )
6		df-dju 7201	. 2 |- ( A ⊔ A ) = ( ( { (/) } X. A ) u. ( { 1o } X. A ) )
7	1, 5, 6	3eqtr4i 2260	1 |- ( 2o X. A ) = ( A ⊔ A )

Syntax hints:   = wceq 1395   u. cun 3195   (/)c0 3491   {csn 3666   {cpr 3667   X. cxp 4716   1oc1o 6553   2oc2o 6554   ⊔ cdju 7200

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-nul 3492  df-pr 3673  df-opab 4145  df-suc 4461  df-xp 4724  df-1o 6560  df-2o 6561  df-dju 7201

This theorem is referenced by: (None)