Theorem elinsn 4245

Description: If the intersection of two classes is a (proper) singleton, then the singleton element is a member of both classes. (Contributed by AV, 30-Dec-2021.)

Assertion

Ref	Expression
elinsn	|- ( ( A e. V /\ ( B i^i C ) = { A } ) -> ( A e. B /\ A e. C ) )

Proof of Theorem elinsn

Step	Hyp	Ref	Expression
1		snidg 4206	. 2 |- ( A e. V -> A e. { A } )
2		eleq2 2690	. . 3 |- ( ( B i^i C ) = { A } -> ( A e. ( B i^i C ) <-> A e. { A } ) )
3		elin 3796	. . . 4 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
4	3	biimpi 206	. . 3 |- ( A e. ( B i^i C ) -> ( A e. B /\ A e. C ) )
5	2, 4	syl6bir 244	. 2 |- ( ( B i^i C ) = { A } -> ( A e. { A } -> ( A e. B /\ A e. C ) ) )
6	1, 5	mpan9 486	1 |- ( ( A e. V /\ ( B i^i C ) = { A } ) -> ( A e. B /\ A e. C ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   {csn 4177

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-in 3581  df-sn 4178

This theorem is referenced by:  frgrncvvdeqlem3  27165  frgrncvvdeqlem6  27168