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Theorem difsnpss 3851
 Description: (B ∖ {A}) is a proper subclass of B if and only if A is a member of B. (Contributed by David Moews, 1-May-2017.)
Assertion
Ref Expression
difsnpss (A B ↔ (B {A}) ⊊ B)

Proof of Theorem difsnpss
StepHypRef Expression
1 notnot 282 . 2 (A B ↔ ¬ ¬ A B)
2 difss 3393 . . . 4 (B {A}) B
32biantrur 492 . . 3 ((B {A}) ≠ B ↔ ((B {A}) B (B {A}) ≠ B))
4 difsnb 3850 . . . 4 A B ↔ (B {A}) = B)
54necon3bbii 2547 . . 3 (¬ ¬ A B ↔ (B {A}) ≠ B)
6 df-pss 3261 . . 3 ((B {A}) ⊊ B ↔ ((B {A}) B (B {A}) ≠ B))
73, 5, 63bitr4i 268 . 2 (¬ ¬ A B ↔ (B {A}) ⊊ B)
81, 7bitri 240 1 (A B ↔ (B {A}) ⊊ B)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ∈ wcel 1710   ≠ wne 2516   ∖ cdif 3206   ⊆ wss 3257   ⊊ wpss 3258  {csn 3737 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-ss 3259  df-pss 3261  df-sn 3741 This theorem is referenced by: (None)
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