Theorem ssdifsn 4737

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 4085	. . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) → 𝐴 ⊆ 𝐵)
2		reldisj 4400	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝐴 ∩ {𝐶}) = ∅ ↔ 𝐴 ⊆ (𝐵 ∖ {𝐶})))
3	2	bicomd 223	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∩ {𝐶}) = ∅))
4	1, 3	biadanii 821	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ {𝐶}) = ∅))
5		disjsn 4661	. . 3 ⊢ ((𝐴 ∩ {𝐶}) = ∅ ↔ ¬ 𝐶 ∈ 𝐴)
6	5	anbi2i 623	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ {𝐶}) = ∅) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴))
7	4, 6	bitri 275	1 ⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ∖ cdif 3894   ∩ cin 3896   ⊆ wss 3897  ∅c0 4280  {csn 4573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-v 3438  df-dif 3900  df-in 3904  df-ss 3914  df-nul 4281  df-sn 4574

This theorem is referenced by:  naddcllem  8591  isdomn6  20629  imadrhmcl  20712  isdrng4  33261  drngmxidl  33442  assafld  33650  logdivsqrle  34663  elsetrecslem  49799