MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snres0 Structured version   Visualization version   GIF version

Theorem snres0 6320
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) = ∅ ↔ ¬ 𝐴𝐶)

Proof of Theorem snres0
StepHypRef Expression
1 relres 6026 . . 3 Rel ({⟨𝐴, 𝐵⟩} ↾ 𝐶)
2 reldm0 5941 . . 3 (Rel ({⟨𝐴, 𝐵⟩} ↾ 𝐶) → (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅))
31, 2ax-mp 5 . 2 (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅)
4 dmres 6032 . . . 4 dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ dom {⟨𝐴, 𝐵⟩})
5 snres0.1 . . . . . 6 𝐵 ∈ V
65dmsnop 6238 . . . . 5 dom {⟨𝐴, 𝐵⟩} = {𝐴}
76ineq2i 4225 . . . 4 (𝐶 ∩ dom {⟨𝐴, 𝐵⟩}) = (𝐶 ∩ {𝐴})
84, 7eqtri 2763 . . 3 dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = (𝐶 ∩ {𝐴})
98eqeq1i 2740 . 2 (dom ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ (𝐶 ∩ {𝐴}) = ∅)
10 disjsn 4716 . 2 ((𝐶 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝐶)
113, 9, 103bitri 297 1 (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ ¬ 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  c0 4339  {csn 4631  cop 4637  dom cdm 5689  cres 5691  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699  df-res 5701
This theorem is referenced by:  noinfbnd2lem1  27790
  Copyright terms: Public domain W3C validator