Theorem xp2dju 7358

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 4750	. 2 |- ( ( { (/) } u. { 1o } ) X. A ) = ( ( { (/) } X. A ) u. ( { 1o } X. A ) )
2		df2o3 6539	. . . 4 |- 2o = { (/) , 1o }
3		df-pr 3650	. . . 4 |- { (/) , 1o } = ( { (/) } u. { 1o } )
4	2, 3	eqtri 2228	. . 3 |- 2o = ( { (/) } u. { 1o } )
5	4	xpeq1i 4713	. 2 |- ( 2o X. A ) = ( ( { (/) } u. { 1o } ) X. A )
6		df-dju 7166	. 2 |- ( A ⊔ A ) = ( ( { (/) } X. A ) u. ( { 1o } X. A ) )
7	1, 5, 6	3eqtr4i 2238	1 |- ( 2o X. A ) = ( A ⊔ A )

Syntax hints:   = wceq 1373   u. cun 3172   (/)c0 3468   {csn 3643   {cpr 3644   X. cxp 4691   1oc1o 6518   2oc2o 6519   ⊔ cdju 7165

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-un 3178  df-nul 3469  df-pr 3650  df-opab 4122  df-suc 4436  df-xp 4699  df-1o 6525  df-2o 6526  df-dju 7166

This theorem is referenced by: (None)