Theorem undisj2 3604

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 3587	. 2 |- (((A i^i B) = (/) /\ (A i^i C) = (/)) <-> ((A i^i B) u. (A i^i C)) = (/))
2		indi 3502	. . 3 |- (A i^i (B u. C)) = ((A i^i B) u. (A i^i C))
3	2	eqeq1i 2360	. 2 |- ((A i^i (B u. C)) = (/) <-> ((A i^i B) u. (A i^i C)) = (/))
4	1, 3	bitr4i 243	1 |- (((A i^i B) = (/) /\ (A i^i C) = (/)) <-> (A i^i (B u. C)) = (/))

Syntax hints:   <-> wb 176   /\ wa 358   = wceq 1642   u. cun 3208   i^i cin 3209  (/)c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by: (None)