Theorem difsnpss 4740

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 316	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ¬ ¬ 𝐴 ∈ 𝐵)
2		difss 4066	. . . 4 ⊢ (𝐵 ∖ {𝐴}) ⊆ 𝐵
3	2	biantrur 535	. . 3 ⊢ ((𝐵 ∖ {𝐴}) ≠ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
4		difsnb 4739	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
5	4	necon3bbii 2981	. . 3 ⊢ (¬ ¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ≠ 𝐵)
6		df-pss 3903	. . 3 ⊢ ((𝐵 ∖ {𝐴}) ⊊ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
7	3, 5, 6	3bitr4i 304	. 2 ⊢ (¬ ¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)
8	1, 7	bitri 276	1 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ∈ wcel 2119   ≠ wne 2934   ∖ cdif 3880   ⊆ wss 3883   ⊊ wpss 3884  {csn 4555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-v 3433  df-dif 3886  df-ss 3900  df-pss 3903  df-sn 4556

This theorem is referenced by:  marypha1lem  9336  infpss  10129  ominf4  10225  mrieqv2d  17596