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Theorem difsnb 4697
 Description: (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 4689. (Contributed by David Moews, 1-May-2017.)
Assertion
Ref Expression
difsnb 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)

Proof of Theorem difsnb
StepHypRef Expression
1 difsn 4689 . 2 𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
2 neldifsnd 4684 . . . . 5 (𝐴𝐵 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))
3 nelne1 3048 . . . . 5 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) → 𝐵 ≠ (𝐵 ∖ {𝐴}))
42, 3mpdan 687 . . . 4 (𝐴𝐵𝐵 ≠ (𝐵 ∖ {𝐴}))
54necomd 3007 . . 3 (𝐴𝐵 → (𝐵 ∖ {𝐴}) ≠ 𝐵)
65necon2bi 2982 . 2 ((𝐵 ∖ {𝐴}) = 𝐵 → ¬ 𝐴𝐵)
71, 6impbii 212 1 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1539   ∈ wcel 2112   ≠ wne 2952   ∖ cdif 3856  {csn 4523 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ne 2953  df-v 3412  df-dif 3862  df-sn 4524 This theorem is referenced by:  difsnpss  4698  incexclem  15240  mrieqv2d  16969  mreexmrid  16973  mreexexlem2d  16975  mreexexlem4d  16977  acsfiindd  17854
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