Theorem undisj2 1740

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 1728	. 2 |- (((A i^i B) = (/) /\ (A i^i C) = (/)) <-> ((A i^i B) u. (A i^i C)) = (/))
2		indi 1676	. . 3 |- (A i^i (B u. C)) = ((A i^i B) u. (A i^i C))
3	2	cleq1i 1108	. 2 |- ((A i^i (B u. C)) = (/) <-> ((A i^i B) u. (A i^i C)) = (/))
4	1, 3	bitr4 154	1 |- (((A i^i B) = (/) /\ (A i^i C) = (/)) <-> (A i^i (B u. C)) = (/))

Syntax hints:   <-> wb 127   /\ wa 196   = wceq 1091   u. cun 1485   i^i cin 1486  (/)c0 1707

This theorem is referenced by:  cdaassen 3725  infxpidmlem11 4943

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708

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