Theorem undisj1 3554

Description: The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)

Assertion

Ref	Expression
undisj1	|- ( ( ( A i^i C ) = (/) /\ ( B i^i C ) = (/) ) <-> ( ( A u. B ) i^i C ) = (/) )

Proof of Theorem undisj1

Step	Hyp	Ref	Expression
1		un00 3543	. 2 |- ( ( ( A i^i C ) = (/) /\ ( B i^i C ) = (/) ) <-> ( ( A i^i C ) u. ( B i^i C ) ) = (/) )
2		indir 3458	. . 3 |- ( ( A u. B ) i^i C ) = ( ( A i^i C ) u. ( B i^i C ) )
3	2	eqeq1i 2239	. 2 |- ( ( ( A u. B ) i^i C ) = (/) <-> ( ( A i^i C ) u. ( B i^i C ) ) = (/) )
4	1, 3	bitr4i 187	1 |- ( ( ( A i^i C ) = (/) /\ ( B i^i C ) = (/) ) <-> ( ( A u. B ) i^i C ) = (/) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   u. cun 3199   i^i cin 3200   (/)c0 3496

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497

This theorem is referenced by:  funtp  5390