Theorem disjtpsn 4251

Description: The disjoint intersection of an unordered triple and a singleton. (Contributed by AV, 14-Nov-2021.)

Assertion

Ref	Expression
disjtpsn	|- ( ( A =/= D /\ B =/= D /\ C =/= D ) -> ( { A , B , C } i^i { D } ) = (/) )

Proof of Theorem disjtpsn

Step	Hyp	Ref	Expression
1		df-tp 4182	. . 3 |- { A , B , C } = ( { A , B } u. { C } )
2	1	ineq1i 3810	. 2 |- ( { A , B , C } i^i { D } ) = ( ( { A , B } u. { C } ) i^i { D } )
3		disjprsn 4250	. . . . 5 |- ( ( A =/= D /\ B =/= D ) -> ( { A , B } i^i { D } ) = (/) )
4	3	3adant3 1081	. . . 4 |- ( ( A =/= D /\ B =/= D /\ C =/= D ) -> ( { A , B } i^i { D } ) = (/) )
5		disjsn2 4247	. . . . 5 |- ( C =/= D -> ( { C } i^i { D } ) = (/) )
6	5	3ad2ant3 1084	. . . 4 |- ( ( A =/= D /\ B =/= D /\ C =/= D ) -> ( { C } i^i { D } ) = (/) )
7	4, 6	jca 554	. . 3 |- ( ( A =/= D /\ B =/= D /\ C =/= D ) -> ( ( { A , B } i^i { D } ) = (/) /\ ( { C } i^i { D } ) = (/) ) )
8		undisj1 4029	. . 3 |- ( ( ( { A , B } i^i { D } ) = (/) /\ ( { C } i^i { D } ) = (/) ) <-> ( ( { A , B } u. { C } ) i^i { D } ) = (/) )
9	7, 8	sylib 208	. 2 |- ( ( A =/= D /\ B =/= D /\ C =/= D ) -> ( ( { A , B } u. { C } ) i^i { D } ) = (/) )
10	2, 9	syl5eq 2668	1 |- ( ( A =/= D /\ B =/= D /\ C =/= D ) -> ( { A , B , C } i^i { D } ) = (/) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   =/= wne 2794   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   {cpr 4179   {ctp 4181

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-3an 1039  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-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-tp 4182

This theorem is referenced by:  disjtp2  4252  cnfldfun  19758