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

Step	Hyp	Ref	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 } )
4	2, 3	eqtri 2133	. . 3 |- 2o = ( { (/) } u. { 1o } )
5	4	xpeq1i 4517	. 2 |- ( 2o X. A ) = ( ( { (/) } u. { 1o } ) X. A )
6		df-dju 6873	. 2 |- ( A ⊔ A ) = ( ( { (/) } X. A ) u. ( { 1o } X. A ) )
7	1, 5, 6	3eqtr4i 2143	1 |- ( 2o X. A ) = ( A ⊔ A )

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)