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Theorem difsnb 4805
Description: (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 4797. (Contributed by David Moews, 1-May-2017.)
Assertion
Ref Expression
difsnb 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)

Proof of Theorem difsnb
StepHypRef Expression
1 difsn 4797 . 2 𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
2 neldifsnd 4792 . . . . 5 (𝐴𝐵 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))
3 nelne1 3038 . . . . 5 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) → 𝐵 ≠ (𝐵 ∖ {𝐴}))
42, 3mpdan 687 . . . 4 (𝐴𝐵𝐵 ≠ (𝐵 ∖ {𝐴}))
54necomd 2995 . . 3 (𝐴𝐵 → (𝐵 ∖ {𝐴}) ≠ 𝐵)
65necon2bi 2970 . 2 ((𝐵 ∖ {𝐴}) = 𝐵 → ¬ 𝐴𝐵)
71, 6impbii 209 1 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1539  wcel 2107  wne 2939  cdif 3947  {csn 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-v 3481  df-dif 3953  df-sn 4626
This theorem is referenced by:  difsnpss  4806  incexclem  15873  mrieqv2d  17683  mreexmrid  17687  mreexexlem2d  17689  mreexexlem4d  17691  acsfiindd  18599
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