Theorem snres0 6262

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 5970	. . 3 ⊢ Rel ({⟨𝐴, 𝐵⟩} ↾ 𝐶)
2		reldm0 5883	. . 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 5977	. . . 4 ⊢ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ dom {⟨𝐴, 𝐵⟩})
5		snres0.1	. . . . . 6 ⊢ 𝐵 ∈ V
6	5	dmsnop 6180	. . . . 5 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}
7	6	ineq2i 4157	. . . 4 ⊢ (𝐶 ∩ dom {⟨𝐴, 𝐵⟩}) = (𝐶 ∩ {𝐴})
8	4, 7	eqtri 2759	. . 3 ⊢ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ {𝐴})
9	8	eqeq1i 2741	. 2 ⊢ (dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ (𝐶 ∩ {𝐴}) = ∅)
10		disjsn 4655	. 2 ⊢ ((𝐶 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐶)
11	3, 9, 10	3bitri 297	1 ⊢ (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ ¬ 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∩ cin 3888  ∅c0 4273  {csn 4567  ⟨cop 4573  dom cdm 5631   ↾ cres 5633  Rel wrel 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-dm 5641  df-res 5643

This theorem is referenced by:  noinfbnd2lem1  27694