Theorem snres0 33724

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 5932	. . 3 ⊢ Rel ({⟨𝐴, 𝐵⟩} ↾ 𝐶)
2		reldm0 5849	. . 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 5925	. . . 4 ⊢ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ dom {⟨𝐴, 𝐵⟩})
5		snres0.1	. . . . . 6 ⊢ 𝐵 ∈ V
6	5	dmsnop 6134	. . . . 5 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}
7	6	ineq2i 4149	. . . 4 ⊢ (𝐶 ∩ dom {⟨𝐴, 𝐵⟩}) = (𝐶 ∩ {𝐴})
8	4, 7	eqtri 2764	. . 3 ⊢ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ {𝐴})
9	8	eqeq1i 2741	. 2 ⊢ (dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ (𝐶 ∩ {𝐴}) = ∅)
10		disjsn 4651	. 2 ⊢ ((𝐶 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐶)
11	3, 9, 10	3bitri 297	1 ⊢ (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ ¬ 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1539   ∈ wcel 2104  Vcvv 3437   ∩ cin 3891  ∅c0 4262  {csn 4565  ⟨cop 4571  dom cdm 5600   ↾ cres 5602  Rel wrel 5605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-xp 5606  df-rel 5607  df-dm 5610  df-res 5612

This theorem is referenced by:  noinfbnd2lem1  33982