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Theorem difsnpss 4732
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 316 . 2 (𝐴𝐵 ↔ ¬ ¬ 𝐴𝐵)
2 difss 4105 . . . 4 (𝐵 ∖ {𝐴}) ⊆ 𝐵
32biantrur 531 . . 3 ((𝐵 ∖ {𝐴}) ≠ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
4 difsnb 4731 . . . 4 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
54necon3bbii 3060 . . 3 (¬ ¬ 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ≠ 𝐵)
6 df-pss 3951 . . 3 ((𝐵 ∖ {𝐴}) ⊊ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
73, 5, 63bitr4i 304 . 2 (¬ ¬ 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)
81, 7bitri 276 1 (𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396  wcel 2105  wne 3013  cdif 3930  wss 3933  wpss 3934  {csn 4557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-pss 3951  df-sn 4558
This theorem is referenced by:  marypha1lem  8885  infpss  9627  ominf4  9722  mrieqv2d  16898
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