Theorem difprsn 2461

Description: Removal of a singleton from an unordered pair.

Assertion

Ref	Expression
difprsn	|- ({A, B} \ {A}) (_ {B}

Proof of Theorem difprsn

Step	Hyp	Ref	Expression
1		pm3.26 319	. . 3 |- ((x = B /\ -. x = A) -> x = B)
2		eldifsn 2458	. . . 4 |- (x e. ({A, B} \ {A}) <-> (x e. {A, B} /\ x =/= A))
3		visset 1809	. . . . . . 7 |- x e. V
4	3	elpr 2420	. . . . . 6 |- (x e. {A, B} <-> (x = A \/ x = B))
5		orcom 246	. . . . . 6 |- ((x = A \/ x = B) <-> (x = B \/ x = A))
6	4, 5	bitr 173	. . . . 5 |- (x e. {A, B} <-> (x = B \/ x = A))
7		df-ne 1584	. . . . 5 |- (x =/= A <-> -. x = A)
8	6, 7	anbi12i 482	. . . 4 |- ((x e. {A, B} /\ x =/= A) <-> ((x = B \/ x = A) /\ -. x = A))
9		pm5.61 447	. . . 4 |- (((x = B \/ x = A) /\ -. x = A) <-> (x = B /\ -. x = A))
10	2, 8, 9	3bitr 177	. . 3 |- (x e. ({A, B} \ {A}) <-> (x = B /\ -. x = A))
11		elsn 2417	. . 3 |- (x e. {B} <-> x = B)
12	1, 10, 11	3imtr4 219	. 2 |- (x e. ({A, B} \ {A}) -> x e. {B})
13	12	ssriv 2065	1 |- ({A, B} \ {A}) (_ {B}

Syntax hints:  -. wn 2   \/ wo 222   /\ wa 223   = wceq 954   e. wcel 956   =/= wne 1582   \ cdif 2040   (_ wss 2043  {csn 2405  {cpr 2406

This theorem is referenced by:  sspr 2471

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-sn 2408  df-pr 2409

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