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Theorem difsnb 4482
Description: (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 4473. (Contributed by David Moews, 1-May-2017.)
Assertion
Ref Expression
difsnb 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)

Proof of Theorem difsnb
StepHypRef Expression
1 difsn 4473 . 2 𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
2 neldifsnd 4468 . . . . 5 (𝐴𝐵 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))
3 nelne1 3028 . . . . 5 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) → 𝐵 ≠ (𝐵 ∖ {𝐴}))
42, 3mpdan 705 . . . 4 (𝐴𝐵𝐵 ≠ (𝐵 ∖ {𝐴}))
54necomd 2987 . . 3 (𝐴𝐵 → (𝐵 ∖ {𝐴}) ≠ 𝐵)
65necon2bi 2962 . 2 ((𝐵 ∖ {𝐴}) = 𝐵 → ¬ 𝐴𝐵)
71, 6impbii 199 1 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1632  wcel 2139  wne 2932  cdif 3712  {csn 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-v 3342  df-dif 3718  df-sn 4322
This theorem is referenced by:  difsnpss  4483  incexclem  14787  mrieqv2d  16521  mreexmrid  16525  mreexexlem2d  16527  mreexexlem4d  16529  acsfiindd  17398
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