Theorem difsn 3846

Description: An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
difsn	⊢ (¬ A ∈ B → (B ∖ {A}) = B)

Proof of Theorem difsn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifsn 3840	. . 3 ⊢ (x ∈ (B ∖ {A}) ↔ (x ∈ B ∧ x ≠ A))
2		simpl 443	. . . 4 ⊢ ((x ∈ B ∧ x ≠ A) → x ∈ B)
3		eleq1 2413	. . . . . . . 8 ⊢ (x = A → (x ∈ B ↔ A ∈ B))
4	3	biimpcd 215	. . . . . . 7 ⊢ (x ∈ B → (x = A → A ∈ B))
5	4	necon3bd 2554	. . . . . 6 ⊢ (x ∈ B → (¬ A ∈ B → x ≠ A))
6	5	com12 27	. . . . 5 ⊢ (¬ A ∈ B → (x ∈ B → x ≠ A))
7	6	ancld 536	. . . 4 ⊢ (¬ A ∈ B → (x ∈ B → (x ∈ B ∧ x ≠ A)))
8	2, 7	impbid2 195	. . 3 ⊢ (¬ A ∈ B → ((x ∈ B ∧ x ≠ A) ↔ x ∈ B))
9	1, 8	syl5bb 248	. 2 ⊢ (¬ A ∈ B → (x ∈ (B ∖ {A}) ↔ x ∈ B))
10	9	eqrdv 2351	1 ⊢ (¬ A ∈ B → (B ∖ {A}) = B)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2517   ∖ cdif 3207  {csn 3738

This proof depends on 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 proof 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-dif 3216  df-sn 3742

This theorem is used by:  difsnb  3851  adj11  3890  nnsucelrlem3  4427  nnsucelr  4429  ssfin  4471