Theorem ressnop0 7022

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 5631	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × V) → 𝐴 ∈ 𝐶)
2		df-res 5602	. . . 4 ⊢ ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ({⟨𝐴, 𝐵⟩} ∩ (𝐶 × V))
3		incom 4140	. . . 4 ⊢ ({⟨𝐴, 𝐵⟩} ∩ (𝐶 × V)) = ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩})
4	2, 3	eqtri 2768	. . 3 ⊢ ({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ((𝐶 × V) ∩ {⟨𝐴, 𝐵⟩})
5		disjsn 4653	. . . 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 2792	. 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 1542   ∈ wcel 2110  Vcvv 3431   ∩ cin 3891  ∅c0 4262  {csn 4567  ⟨cop 4573   × cxp 5588   ↾ cres 5592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5142  df-xp 5596  df-res 5602

This theorem is referenced by:  fvunsn  7048  fsnunres  7057  frrlem12  8104  wfrlem14OLD  8144  ex-res  28801