Theorem undisj2 3496

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 3484	. 2 |- ( ( ( A i^i B ) = (/) /\ ( A i^i C ) = (/) ) <-> ( ( A i^i B ) u. ( A i^i C ) ) = (/) )
2		indi 3397	. . 3 |- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) )
3	2	eqeq1i 2197	. 2 |- ( ( A i^i ( B u. C ) ) = (/) <-> ( ( A i^i B ) u. ( A i^i C ) ) = (/) )
4	1, 3	bitr4i 187	1 |- ( ( ( A i^i B ) = (/) /\ ( A i^i C ) = (/) ) <-> ( A i^i ( B u. C ) ) = (/) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   u. cun 3142   i^i cin 3143   (/)c0 3437

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438

This theorem is referenced by: (None)