Theorem ressnop0 7142

Description: If 𝐴 is not in 𝐶, then the restriction of a singleton of ⟨𝐴, 𝐵⟩ to 𝐶 is null. (Contributed by Scott Fenton, 15-Apr-2011.)

Assertion

Ref	Expression
ressnop0	⊢ (¬ 𝐴 ∈ 𝐶 → ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅)

Proof of Theorem ressnop0

Step	Hyp	Ref	Expression
1		opelxp1 5696	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × V) → 𝐴 ∈ 𝐶)
2		df-res 5666	. . . 4 ⊢ ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ({⟨𝐴, 𝐵⟩} ∩ (𝐶 × V))
3		incom 4184	. . . 4 ⊢ ({⟨𝐴, 𝐵⟩} ∩ (𝐶 × V)) = ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩})
4	2, 3	eqtri 2758	. . 3 ⊢ ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩})
5		disjsn 4687	. . . 4 ⊢ (((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩}) = ∅ ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V))
6	5	biimpri 228	. . 3 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V) → ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩}) = ∅)
7	4, 6	eqtrid 2782	. 2 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V) → ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅)
8	1, 7	nsyl5 159	1 ⊢ (¬ 𝐴 ∈ 𝐶 → ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ∩ cin 3925  ∅c0 4308  {csn 4601  ⟨cop 4607   × cxp 5652   ↾ cres 5656

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-opab 5182  df-xp 5660  df-res 5666

This theorem is referenced by:  fvunsn  7170  fsnunres  7179  frrlem12  8294  wfrlem14OLD  8334  ex-res  30368