Theorem undisj1 3279

Description: The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)

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 3263	. 2 |- ( ( ( A i^i C ) = (/) /\ ( B i^i C ) = (/) ) <-> ( ( A i^i C ) u. ( B i^i C ) ) = (/) )
2		indir 3186	. . 3 |- ( ( A u. B ) i^i C ) = ( ( A i^i C ) u. ( B i^i C ) )
3	2	eqeq1i 2047	. 2 |- ( ( ( A u. B ) i^i C ) = (/) <-> ( ( A i^i C ) u. ( B i^i C ) ) = (/) )
4	1, 3	bitr4i 176	1 |- ( ( ( A i^i C ) = (/) /\ ( B i^i C ) = (/) ) <-> ( ( A u. B ) i^i C ) = (/) )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   u. cun 2915   i^i cin 2916   (/)c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225

This theorem is referenced by:  funtp  4952