Theorem undisj2 2926

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 2907	. 2 |- (((A i^i B) = (/) /\ (A i^i C) = (/)) <-> ((A i^i B) u. (A i^i C)) = (/))
2		indi 2838	. . 3 |- (A i^i (B u. C)) = ((A i^i B) u. (A i^i C))
3	2	eqeq1i 1891	. 2 |- ((A i^i (B u. C)) = (/) <-> ((A i^i B) u. (A i^i C)) = (/))
4	1, 3	bitr4i 193	1 |- (((A i^i B) = (/) /\ (A i^i C) = (/)) <-> (A i^i (B u. C)) = (/))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   u. cun 2591   i^i cin 2592  (/)c0 2875

This theorem is referenced by:  cdaassen 6080  renfdisjOLD 6713  infxpidmlem11 8831

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876

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