Theorem difsn 4802

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 4791	. . 3 ⊢ (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴))
2		simpl 484	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴) → 𝑥 ∈ 𝐵)
3		nelelne 3042	. . . . 5 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ≠ 𝐴))
4	3	ancld 552	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 𝐴)))
5	2, 4	impbid2 225	. . 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 2731	1 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ≠ wne 2941   ∖ cdif 3946  {csn 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-v 3477  df-dif 3952  df-sn 4630

This theorem is referenced by:  difsnb  4810  difsnexi  7748  domdifsn  9054  domunsncan  9072  frfi  9288  infdifsn  9652  dfn2  12485  hashgt23el  14384  clslp  22652  xrge00  32187  lindsadd  36481  lindsenlbs  36483  poimirlem2  36490  poimirlem4  36492  poimirlem6  36494  poimirlem7  36495  poimirlem8  36496  poimirlem19  36507  poimirlem23  36511  supxrmnf2  44143  infxrpnf2  44173  dvmptfprodlem  44660  hoiprodp1  45304