Theorem snres0 6253

Description: Condition for restriction of a singleton to be empty. (Contributed by Scott Fenton, 9-Aug-2024.)

Hypothesis

Ref	Expression
snres0.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
snres0	⊢ (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ ¬ 𝐴 ∈ 𝐶)

Proof of Theorem snres0

Step	Hyp	Ref	Expression
1		relres 5961	. . 3 ⊢ Rel ({⟨𝐴, 𝐵⟩} ↾ 𝐶)
2		reldm0 5874	. . 3 ⊢ (Rel ({⟨𝐴, 𝐵⟩} ↾ 𝐶) → (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅))
3	1, 2	ax-mp 5	. 2 ⊢ (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅)
4		dmres 5968	. . . 4 ⊢ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ dom {⟨𝐴, 𝐵⟩})
5		snres0.1	. . . . . 6 ⊢ 𝐵 ∈ V
6	5	dmsnop 6171	. . . . 5 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}
7	6	ineq2i 4166	. . . 4 ⊢ (𝐶 ∩ dom {⟨𝐴, 𝐵⟩}) = (𝐶 ∩ {𝐴})
8	4, 7	eqtri 2756	. . 3 ⊢ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ {𝐴})
9	8	eqeq1i 2738	. 2 ⊢ (dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ (𝐶 ∩ {𝐴}) = ∅)
10		disjsn 4665	. 2 ⊢ ((𝐶 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐶)
11	3, 9, 10	3bitri 297	1 ⊢ (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ ¬ 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ∩ cin 3897  ∅c0 4282  {csn 4577  ⟨cop 4583  dom cdm 5621   ↾ cres 5623  Rel wrel 5626

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 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-dm 5631  df-res 5633

This theorem is referenced by:  noinfbnd2lem1  27689