Theorem difsnb 4809

Description: (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 4801. (Contributed by David Moews, 1-May-2017.)

Assertion

Ref	Expression
difsnb	⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)

Proof of Theorem difsnb

Step	Hyp	Ref	Expression
1		difsn 4801	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
2		neldifsnd 4796	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))
3		nelne1 3040	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) → 𝐵 ≠ (𝐵 ∖ {𝐴}))
4	2, 3	mpdan 686	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ (𝐵 ∖ {𝐴}))
5	4	necomd 2997	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) ≠ 𝐵)
6	5	necon2bi 2972	. 2 ⊢ ((𝐵 ∖ {𝐴}) = 𝐵 → ¬ 𝐴 ∈ 𝐵)
7	1, 6	impbii 208	1 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1542   ∈ wcel 2107   ≠ wne 2941   ∖ cdif 3945  {csn 4628

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 3951  df-sn 4629

This theorem is referenced by:  difsnpss  4810  incexclem  15779  mrieqv2d  17580  mreexmrid  17584  mreexexlem2d  17586  mreexexlem4d  17588  acsfiindd  18503