Theorem eldifeldifsn 4342

Description: An element of a difference set is an element of the difference with a singleton. (Contributed by AV, 2-Jan-2022.)

Assertion

Ref	Expression
eldifeldifsn	|- ( ( X e. A /\ Y e. ( B \ A ) ) -> Y e. ( B \ { X } ) )

Proof of Theorem eldifeldifsn

Step	Hyp	Ref	Expression
1		snssi 4339	. . 3 |- ( X e. A -> { X } C_ A )
2	1	sscond 3747	. 2 |- ( X e. A -> ( B \ A ) C_ ( B \ { X } ) )
3	2	sselda 3603	1 |- ( ( X e. A /\ Y e. ( B \ A ) ) -> Y e. ( B \ { X } ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   \ cdif 3571   {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-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-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-sn 4178

This theorem is referenced by: (None)