Theorem difsn 3781

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 3771	. . 3 ⊢ (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴))
2		simpl 109	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴) → 𝑥 ∈ 𝐵)
3		eleq1 2270	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
4	3	biimpcd 159	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
5	4	necon3bd 2421	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐵 → 𝑥 ≠ 𝐴))
6	5	com12 30	. . . . 5 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ≠ 𝐴))
7	6	ancld 325	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴)))
8	2, 7	impbid2 143	. . 3 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴) ↔ 𝑥 ∈ 𝐵))
9	1, 8	bitrid 192	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ 𝑥 ∈ 𝐵))
10	9	eqrdv 2205	1 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2178   ≠ wne 2378   ∖ cdif 3171  {csn 3643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-v 2778  df-dif 3176  df-sn 3649

This theorem is referenced by:  difsnb  3787  fisseneq  7057  dfn2  9343