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Theorem difsnb 3850
 Description: (B ∖ {A}) equals B if and only if A is not a member of B. Generalization of difsn 3845. (Contributed by David Moews, 1-May-2017.)
Assertion
Ref Expression
difsnb A B ↔ (B {A}) = B)

Proof of Theorem difsnb
StepHypRef Expression
1 difsn 3845 . 2 A B → (B {A}) = B)
2 neldifsnd 3842 . . . . 5 (A B → ¬ A (B {A}))
3 nelne1 2605 . . . . 5 ((A B ¬ A (B {A})) → B ≠ (B {A}))
42, 3mpdan 649 . . . 4 (A BB ≠ (B {A}))
54necomd 2599 . . 3 (A B → (B {A}) ≠ B)
65necon2bi 2562 . 2 ((B {A}) = B → ¬ A B)
71, 6impbii 180 1 A B ↔ (B {A}) = B)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   = wceq 1642   ∈ wcel 1710   ≠ wne 2516   ∖ cdif 3206  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-sn 3741 This theorem is referenced by:  difsnpss  3851
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