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 ∈ A → ((A ∪ {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 ∈ ((A ∪ {B}) ∖ {B}) ↔ (x ∈ (A ∪ {B}) ∧ x ≠ B))
2		elun 3221	. . . . . . 7 ⊢ (x ∈ (A ∪ {B}) ↔ (x ∈ A ∨ x ∈ {B}))
3		elsn 3749	. . . . . . . 8 ⊢ (x ∈ {B} ↔ x = B)
4	3	orbi2i 505	. . . . . . 7 ⊢ ((x ∈ A ∨ x ∈ {B}) ↔ (x ∈ A ∨ x = B))
5	2, 4	bitri 240	. . . . . 6 ⊢ (x ∈ (A ∪ {B}) ↔ (x ∈ A ∨ x = B))
6		df-ne 2519	. . . . . 6 ⊢ (x ≠ B ↔ ¬ x = B)
7	5, 6	anbi12i 678	. . . . 5 ⊢ ((x ∈ (A ∪ {B}) ∧ x ≠ B) ↔ ((x ∈ A ∨ x = B) ∧ ¬ x = B))
8		pm5.61 693	. . . . 5 ⊢ (((x ∈ A ∨ x = B) ∧ ¬ x = B) ↔ (x ∈ A ∧ ¬ x = B))
9	1, 7, 8	3bitri 262	. . . 4 ⊢ (x ∈ ((A ∪ {B}) ∖ {B}) ↔ (x ∈ A ∧ ¬ x = B))
10		ancom 437	. . . 4 ⊢ ((x ∈ A ∧ ¬ x = B) ↔ (¬ x = B ∧ x ∈ A))
11	9, 10	bitri 240	. . 3 ⊢ (x ∈ ((A ∪ {B}) ∖ {B}) ↔ (¬ x = B ∧ x ∈ A))
12		eleq1 2413	. . . . . . . 8 ⊢ (x = B → (x ∈ A ↔ B ∈ A))
13	12	biimpcd 215	. . . . . . 7 ⊢ (x ∈ A → (x = B → B ∈ A))
14	13	con3d 125	. . . . . 6 ⊢ (x ∈ A → (¬ B ∈ A → ¬ x = B))
15	14	com12 27	. . . . 5 ⊢ (¬ B ∈ A → (x ∈ A → ¬ x = B))
16	15	pm4.71rd 616	. . . 4 ⊢ (¬ B ∈ A → (x ∈ A ↔ (¬ x = B ∧ x ∈ A)))
17	16	bicomd 192	. . 3 ⊢ (¬ B ∈ A → ((¬ x = B ∧ x ∈ A) ↔ x ∈ A))
18	11, 17	syl5bb 248	. 2 ⊢ (¬ B ∈ A → (x ∈ ((A ∪ {B}) ∖ {B}) ↔ x ∈ A))
19	18	eqrdv 2351	1 ⊢ (¬ B ∈ A → ((A ∪ {B}) ∖ {B}) = A)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2517   ∖ cdif 3207   ∪ 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