Theorem difsn 3665

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 3658	. . 3 ⊢ (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴))
2		simpl 108	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴) → 𝑥 ∈ 𝐵)
3		eleq1 2203	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
4	3	biimpcd 158	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
5	4	necon3bd 2352	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐵 → 𝑥 ≠ 𝐴))
6	5	com12 30	. . . . 5 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ≠ 𝐴))
7	6	ancld 323	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴)))
8	2, 7	impbid2 142	. . 3 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴) ↔ 𝑥 ∈ 𝐵))
9	1, 8	syl5bb 191	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ 𝑥 ∈ 𝐵))
10	9	eqrdv 2138	1 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481   ≠ wne 2309   ∖ cdif 3073  {csn 3532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2691  df-dif 3078  df-sn 3538

This theorem is referenced by:  difsnb  3671  fisseneq  6828  dfn2  9014