Theorem undisj1 3472

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 3461	. 2 |- ( ( ( A i^i C ) = (/) /\ ( B i^i C ) = (/) ) <-> ( ( A i^i C ) u. ( B i^i C ) ) = (/) )
2		indir 3376	. . 3 |- ( ( A u. B ) i^i C ) = ( ( A i^i C ) u. ( B i^i C ) )
3	2	eqeq1i 2178	. 2 |- ( ( ( A u. B ) i^i C ) = (/) <-> ( ( A i^i C ) u. ( B i^i C ) ) = (/) )
4	1, 3	bitr4i 186	1 |- ( ( ( A i^i C ) = (/) /\ ( B i^i C ) = (/) ) <-> ( ( A u. B ) i^i C ) = (/) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1348   u. cun 3119   i^i cin 3120   (/)c0 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415

This theorem is referenced by:  funtp  5251