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Theorem difsnpss 4805
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 315 . 2 (𝐴𝐵 ↔ ¬ ¬ 𝐴𝐵)
2 difss 4126 . . . 4 (𝐵 ∖ {𝐴}) ⊆ 𝐵
32biantrur 530 . . 3 ((𝐵 ∖ {𝐴}) ≠ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
4 difsnb 4804 . . . 4 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
54necon3bbii 2982 . . 3 (¬ ¬ 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ≠ 𝐵)
6 df-pss 3962 . . 3 ((𝐵 ∖ {𝐴}) ⊊ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
73, 5, 63bitr4i 303 . 2 (¬ ¬ 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)
81, 7bitri 275 1 (𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 395  wcel 2098  wne 2934  cdif 3940  wss 3943  wpss 3944  {csn 4623
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 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-v 3470  df-dif 3946  df-in 3950  df-ss 3960  df-pss 3962  df-sn 4624
This theorem is referenced by:  marypha1lem  9430  infpss  10214  ominf4  10309  mrieqv2d  17592
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