Theorem undisj2 2325

Description: The union of disjoint classes is disjoint.

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 2310	. 2 |- (((A i^i B) = (/) /\ (A i^i C) = (/)) <-> ((A i^i B) u. (A i^i C)) = (/))
2		indi 2254	. . 3 |- (A i^i (B u. C)) = ((A i^i B) u. (A i^i C))
3	2	eqeq1i 1485	. 2 |- ((A i^i (B u. C)) = (/) <-> ((A i^i B) u. (A i^i C)) = (/))
4	1, 3	bitr4 176	1 |- (((A i^i B) = (/) /\ (A i^i C) = (/)) <-> (A i^i (B u. C)) = (/))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   u. cun 2048   i^i cin 2049  (/)c0 2283

This theorem is referenced by:  cdaassen 4942  renfdisj 5551  infxpidmlem11 7563

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284

Copyright terms: Public domain