Theorem snres0 6329

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 6035	. . 3 ⊢ Rel ({⟨𝐴, 𝐵⟩} ↾ 𝐶)
2		reldm0 5952	. . 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 6041	. . . 4 ⊢ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ dom {⟨𝐴, 𝐵⟩})
5		snres0.1	. . . . . 6 ⊢ 𝐵 ∈ V
6	5	dmsnop 6247	. . . . 5 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}
7	6	ineq2i 4238	. . . 4 ⊢ (𝐶 ∩ dom {⟨𝐴, 𝐵⟩}) = (𝐶 ∩ {𝐴})
8	4, 7	eqtri 2768	. . 3 ⊢ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ {𝐴})
9	8	eqeq1i 2745	. 2 ⊢ (dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ (𝐶 ∩ {𝐴}) = ∅)
10		disjsn 4736	. 2 ⊢ ((𝐶 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐶)
11	3, 9, 10	3bitri 297	1 ⊢ (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ ¬ 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∩ cin 3975  ∅c0 4352  {csn 4648  ⟨cop 4654  dom cdm 5700   ↾ cres 5702  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-dm 5710  df-res 5712

This theorem is referenced by:  noinfbnd2lem1  27793