Theorem difsnpss 4279

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	⊢ (𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)

Proof of Theorem difsnpss

Step	Hyp	Ref	Expression
1		notnotb 303	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ¬ ¬ 𝐴 ∈ 𝐵)
2		difss 3699	. . . 4 ⊢ (𝐵 ∖ {𝐴}) ⊆ 𝐵
3	2	biantrur 526	. . 3 ⊢ ((𝐵 ∖ {𝐴}) ≠ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
4		difsnb 4278	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
5	4	necon3bbii 2829	. . 3 ⊢ (¬ ¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ≠ 𝐵)
6		df-pss 3556	. . 3 ⊢ ((𝐵 ∖ {𝐴}) ⊊ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
7	3, 5, 6	3bitr4i 291	. 2 ⊢ (¬ ¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)
8	1, 7	bitri 263	1 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)

Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537   ⊆ wss 3540   ⊊ wpss 3541  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-pss 3556  df-sn 4126

This theorem is referenced by:  marypha1lem  8222  infpss  8922  ominf4  9017  mrieqv2d  16122