Theorem ressnop0 7025

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 5630	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × V) → 𝐴 ∈ 𝐶)
2		df-res 5601	. . . 4 ⊢ ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ({⟨𝐴, 𝐵⟩} ∩ (𝐶 × V))
3		incom 4135	. . . 4 ⊢ ({⟨𝐴, 𝐵⟩} ∩ (𝐶 × V)) = ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩})
4	2, 3	eqtri 2766	. . 3 ⊢ ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩})
5		disjsn 4647	. . . 4 ⊢ (((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩}) = ∅ ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V))
6	5	biimpri 227	. . 3 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ (𝐶 × V) → ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩}) = ∅)
7	4, 6	eqtrid 2790	. 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 1539   ∈ wcel 2106  Vcvv 3432   ∩ cin 3886  ∅c0 4256  {csn 4561  ⟨cop 4567   × cxp 5587   ↾ cres 5591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-xp 5595  df-res 5601

This theorem is referenced by:  fvunsn  7051  fsnunres  7060  frrlem12  8113  wfrlem14OLD  8153  ex-res  28805