Theorem ssdifsn 4721

Description: Subset of a set with an element removed. (Contributed by Emmett Weisz, 7-Jul-2021.) (Proof shortened by JJ, 31-May-2022.)

Assertion

Ref	Expression
ssdifsn	⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴))

Proof of Theorem ssdifsn

Step	Hyp	Ref	Expression
1		difss2 4068	. . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) → 𝐴 ⊆ 𝐵)
2		reldisj 4385	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐴 ∩ {𝐶}) = ∅ ↔ 𝐴 ⊆ (𝐵 ∖ {𝐶})))
3	2	bicomd 222	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∩ {𝐶}) = ∅))
4	1, 3	biadanii 819	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ {𝐶}) = ∅))
5		disjsn 4647	. . 3 ⊢ ((𝐴 ∩ {𝐶}) = ∅ ↔ ¬ 𝐶 ∈ 𝐴)
6	5	anbi2i 623	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ {𝐶}) = ∅) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴))
7	4, 6	bitri 274	1 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {csn 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562

This theorem is referenced by:  logdivsqrle  32630  naddcllem  33831  elsetrecslem  46404