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Theorem difsnpss 4806
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 4135 . . . 4 (𝐵 ∖ {𝐴}) ⊆ 𝐵
32biantrur 530 . . 3 ((𝐵 ∖ {𝐴}) ≠ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
4 difsnb 4805 . . . 4 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
54necon3bbii 2987 . . 3 (¬ ¬ 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ≠ 𝐵)
6 df-pss 3970 . . 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 206  wa 395  wcel 2107  wne 2939  cdif 3947  wss 3950  wpss 3951  {csn 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-v 3481  df-dif 3953  df-ss 3967  df-pss 3970  df-sn 4626
This theorem is referenced by:  marypha1lem  9474  infpss  10257  ominf4  10353  mrieqv2d  17683
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