Theorem undisj1 2320

Description: The union of disjoint classes is disjoint.

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 2306	. 2 |- (((A i^i C) = (/) /\ (B i^i C) = (/)) <-> ((A i^i C) u. (B i^i C)) = (/))
2		indir 2253	. . 3 |- ((A u. B) i^i C) = ((A i^i C) u. (B i^i C))
3	2	eqeq1i 1482	. 2 |- (((A u. B) i^i C) = (/) <-> ((A i^i C) u. (B i^i C)) = (/))
4	1, 3	bitr4 176	1 |- (((A i^i C) = (/) /\ (B i^i C) = (/)) <-> ((A u. B) i^i C) = (/))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   u. cun 2045   i^i cin 2046  (/)c0 2280

This theorem is referenced by:  cdaassen 4930

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281

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