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Theorem difsnb 3506
 Description: (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 3501. (Contributed by David Moews, 1-May-2017.)
Assertion
Ref Expression
difsnb 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)

Proof of Theorem difsnb
StepHypRef Expression
1 difsn 3501 . 2 𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
2 neldifsnd 3498 . . . . 5 (𝐴𝐵 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))
3 nelne1 2295 . . . . 5 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) → 𝐵 ≠ (𝐵 ∖ {𝐴}))
42, 3mpdan 398 . . . 4 (𝐴𝐵𝐵 ≠ (𝐵 ∖ {𝐴}))
54necomd 2291 . . 3 (𝐴𝐵 → (𝐵 ∖ {𝐴}) ≠ 𝐵)
65necon2bi 2260 . 2 ((𝐵 ∖ {𝐴}) = 𝐵 → ¬ 𝐴𝐵)
71, 6impbii 117 1 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 98   = wceq 1243   ∈ wcel 1393   ≠ wne 2204   ∖ cdif 2914  {csn 3375 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-sn 3381 This theorem is referenced by:  difsnpssim  3507
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