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Theorem difsnpss 4307
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 304 . 2 (𝐴𝐵 ↔ ¬ ¬ 𝐴𝐵)
2 difss 3715 . . . 4 (𝐵 ∖ {𝐴}) ⊆ 𝐵
32biantrur 527 . . 3 ((𝐵 ∖ {𝐴}) ≠ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
4 difsnb 4306 . . . 4 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
54necon3bbii 2837 . . 3 (¬ ¬ 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ≠ 𝐵)
6 df-pss 3571 . . 3 ((𝐵 ∖ {𝐴}) ⊊ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
73, 5, 63bitr4i 292 . 2 (¬ ¬ 𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)
81, 7bitri 264 1 (𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384  wcel 1987  wne 2790  cdif 3552  wss 3555  wpss 3556  {csn 4148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-pss 3571  df-sn 4149
This theorem is referenced by:  marypha1lem  8283  infpss  8983  ominf4  9078  mrieqv2d  16220
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