Theorem undisj1 3451

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 3440	. 2 |- ( ( ( A i^i C ) = (/) /\ ( B i^i C ) = (/) ) <-> ( ( A i^i C ) u. ( B i^i C ) ) = (/) )
2		indir 3356	. . 3 |- ( ( A u. B ) i^i C ) = ( ( A i^i C ) u. ( B i^i C ) )
3	2	eqeq1i 2165	. 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 1335   u. cun 3100   i^i cin 3101   (/)c0 3394

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395

This theorem is referenced by:  funtp  5222