Theorem undisj2 3519

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 3507	. 2 |- ( ( ( A i^i B ) = (/) /\ ( A i^i C ) = (/) ) <-> ( ( A i^i B ) u. ( A i^i C ) ) = (/) )
2		indi 3420	. . 3 |- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) )
3	2	eqeq1i 2213	. 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 1373   u. cun 3164   i^i cin 3165   (/)c0 3460

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461

This theorem is referenced by: (None)