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Theorem difsnb 4525
Description: (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 4517. (Contributed by David Moews, 1-May-2017.)
Assertion
Ref Expression
difsnb 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)

Proof of Theorem difsnb
StepHypRef Expression
1 difsn 4517 . 2 𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
2 neldifsnd 4512 . . . . 5 (𝐴𝐵 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))
3 nelne1 3067 . . . . 5 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) → 𝐵 ≠ (𝐵 ∖ {𝐴}))
42, 3mpdan 679 . . . 4 (𝐴𝐵𝐵 ≠ (𝐵 ∖ {𝐴}))
54necomd 3026 . . 3 (𝐴𝐵 → (𝐵 ∖ {𝐴}) ≠ 𝐵)
65necon2bi 3001 . 2 ((𝐵 ∖ {𝐴}) = 𝐵 → ¬ 𝐴𝐵)
71, 6impbii 201 1 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198   = wceq 1653  wcel 2157  wne 2971  cdif 3766  {csn 4368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-v 3387  df-dif 3772  df-sn 4369
This theorem is referenced by:  difsnpss  4526  incexclem  14906  mrieqv2d  16614  mreexmrid  16618  mreexexlem2d  16620  mreexexlem4d  16622  acsfiindd  17492
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