Theorem undisj2 3302

Description: The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)

Assertion

Ref	Expression
undisj2	|- ( ( ( A i^i B ) = (/) /\ ( A i^i C ) = (/) ) <-> ( A i^i ( B u. C ) ) = (/) )

Proof of Theorem undisj2

Step	Hyp	Ref	Expression
1		un00 3290	. 2 |- ( ( ( A i^i B ) = (/) /\ ( A i^i C ) = (/) ) <-> ( ( A i^i B ) u. ( A i^i C ) ) = (/) )
2		indi 3211	. . 3 |- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) )
3	2	eqeq1i 2088	. 2 |- ( ( A i^i ( B u. C ) ) = (/) <-> ( ( A i^i B ) u. ( A i^i C ) ) = (/) )
4	1, 3	bitr4i 185	1 |- ( ( ( A i^i B ) = (/) /\ ( A i^i C ) = (/) ) <-> ( A i^i ( B u. C ) ) = (/) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   u. cun 2971   i^i cin 2972   (/)c0 3251

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252

This theorem is referenced by: (None)