Theorem disjprsn 4250

Description: The disjoint intersection of an unordered pair and a singleton. (Contributed by AV, 23-Jan-2021.)

Assertion

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

Proof of Theorem disjprsn

Step	Hyp	Ref	Expression
1		dfsn2 4190	. . 3 |- { C } = { C , C }
2	1	ineq2i 3811	. 2 |- ( { A , B } i^i { C } ) = ( { A , B } i^i { C , C } )
3		disjpr2 4248	. . 3 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= C /\ B =/= C ) ) -> ( { A , B } i^i { C , C } ) = (/) )
4	3	anidms 677	. 2 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } i^i { C , C } ) = (/) )
5	2, 4	syl5eq 2668	1 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } i^i { C } ) = (/) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794   i^i cin 3573   (/)c0 3915   {csn 4177   {cpr 4179

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-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

This theorem is referenced by:  disjtpsn  4251  disjtp2  4252  diftpsn3  4332  funtpg  5942  funcnvtp  5951  prodtp  29573