Theorem nnsucelrlem2 4426

Description: Lemma for nnsucelr 4429. Subtracting a non-element from a set adjoined with the non-element retrieves the original set. (Contributed by SF, 15-Jan-2015.)

Assertion

Ref	Expression
nnsucelrlem2	|- (-. B e. A -> ((A u. {B}) \ {B}) = A)

Proof of Theorem nnsucelrlem2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifsn 3840	. . . . 5 |- (x e. ((A u. {B}) \ {B}) <-> (x e. (A u. {B}) /\ x =/= B))
2		elun 3221	. . . . . . 7 |- (x e. (A u. {B}) <-> (x e. A \/ x e. {B}))
3		elsn 3749	. . . . . . . 8 |- (x e. {B} <-> x = B)
4	3	orbi2i 505	. . . . . . 7 |- ((x e. A \/ x e. {B}) <-> (x e. A \/ x = B))
5	2, 4	bitri 240	. . . . . 6 |- (x e. (A u. {B}) <-> (x e. A \/ x = B))
6		df-ne 2519	. . . . . 6 |- (x =/= B <-> -. x = B)
7	5, 6	anbi12i 678	. . . . 5 |- ((x e. (A u. {B}) /\ x =/= B) <-> ((x e. A \/ x = B) /\ -. x = B))
8		pm5.61 693	. . . . 5 |- (((x e. A \/ x = B) /\ -. x = B) <-> (x e. A /\ -. x = B))
9	1, 7, 8	3bitri 262	. . . 4 |- (x e. ((A u. {B}) \ {B}) <-> (x e. A /\ -. x = B))
10		ancom 437	. . . 4 |- ((x e. A /\ -. x = B) <-> (-. x = B /\ x e. A))
11	9, 10	bitri 240	. . 3 |- (x e. ((A u. {B}) \ {B}) <-> (-. x = B /\ x e. A))
12		eleq1 2413	. . . . . . . 8 |- (x = B -> (x e. A <-> B e. A))
13	12	biimpcd 215	. . . . . . 7 |- (x e. A -> (x = B -> B e. A))
14	13	con3d 125	. . . . . 6 |- (x e. A -> (-. B e. A -> -. x = B))
15	14	com12 27	. . . . 5 |- (-. B e. A -> (x e. A -> -. x = B))
16	15	pm4.71rd 616	. . . 4 |- (-. B e. A -> (x e. A <-> (-. x = B /\ x e. A)))
17	16	bicomd 192	. . 3 |- (-. B e. A -> ((-. x = B /\ x e. A) <-> x e. A))
18	11, 17	syl5bb 248	. 2 |- (-. B e. A -> (x e. ((A u. {B}) \ {B}) <-> x e. A))
19	18	eqrdv 2351	1 |- (-. B e. A -> ((A u. {B}) \ {B}) = A)

Syntax hints:  -. wn 3   -> wi 4   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2517   \ cdif 3207   u. cun 3208  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-sn 3742

This theorem is referenced by:  nnsucelr  4429  enadj  6061