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Theorem difsnpss 4815
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 (𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)

Proof of Theorem difsnpss
StepHypRef Expression
1 notnotb 314 . 2 (𝐴𝐵 ↔ ¬ ¬ 𝐴𝐵)
2 difss 4132 . . . 4 (𝐵 ∖ {𝐴}) ⊆ 𝐵
32biantrur 529 . . 3 ((𝐵 ∖ {𝐴}) ≠ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
4 difsnb 4814 . . . 4 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
54necon3bbii 2985 . . 3 (¬ ¬ 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ≠ 𝐵)
6 df-pss 3968 . . 3 ((𝐵 ∖ {𝐴}) ⊊ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
73, 5, 63bitr4i 302 . 2 (¬ ¬ 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)
81, 7bitri 274 1 (𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 394  wcel 2098  wne 2937  cdif 3946  wss 3949  wpss 3950  {csn 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-v 3475  df-dif 3952  df-in 3956  df-ss 3966  df-pss 3968  df-sn 4633
This theorem is referenced by:  marypha1lem  9466  infpss  10250  ominf4  10345  mrieqv2d  17628
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