Theorem difsn 4754

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 4742	. . 3 ⊢ (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴))
2		simpl 482	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴) → 𝑥 ∈ 𝐵)
3		nelelne 3031	. . . . 5 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ≠ 𝐴))
4	3	ancld 550	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴)))
5	2, 4	impbid2 226	. . 3 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴) ↔ 𝑥 ∈ 𝐵))
6	1, 5	bitrid 283	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ 𝑥 ∈ 𝐵))
7	6	eqrdv 2734	1 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932   ∖ cdif 3898  {csn 4580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-v 3442  df-dif 3904  df-sn 4581

This theorem is referenced by:  difsnb  4762  difsnexi  7706  domdifsn  8988  domunsncan  9005  frfi  9185  infdifsn  9566  dfn2  12414  hashgt23el  14347  chnccat  18549  clslp  23092  xrge00  33096  lindsadd  37814  lindsenlbs  37816  poimirlem2  37823  poimirlem4  37825  poimirlem6  37827  poimirlem7  37828  poimirlem8  37829  poimirlem19  37840  poimirlem23  37844  supxrmnf2  45687  infxrpnf2  45717  dvmptfprodlem  46198  hoiprodp1  46842