Theorem undisj2 4030

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 4011	. 2 |- ( ( ( A i^i B ) = (/) /\ ( A i^i C ) = (/) ) <-> ( ( A i^i B ) u. ( A i^i C ) ) = (/) )
2		indi 3873	. . 3 |- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) )
3	2	eqeq1i 2627	. 2 |- ( ( A i^i ( B u. C ) ) = (/) <-> ( ( A i^i B ) u. ( A i^i C ) ) = (/) )
4	1, 3	bitr4i 267	1 |- ( ( ( A i^i B ) = (/) /\ ( A i^i C ) = (/) ) <-> ( A i^i ( B u. C ) ) = (/) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   u. cun 3572   i^i cin 3573   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  disjtp2  4252  f1oun2prg  13662  cnfldfun  19758