Theorem iin0 5335

Description: An indexed intersection of the empty set, with a nonempty index set, is empty. (Contributed by NM, 20-Oct-2005.)

Assertion

Ref	Expression
iin0	⊢ (𝐴 ≠ ∅ ↔ ∩ 𝑥 ∈ 𝐴 ∅ = ∅)

Distinct variable group:   𝑥,𝐴

Proof of Theorem iin0

Step	Hyp	Ref	Expression
1		iinconst 4969	. 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∅)
2		0ex 5272	. . . . . 6 ⊢ ∅ ∈ V
3	2	n0ii 4296	. . . . 5 ⊢ ¬ V = ∅
4		0iin 5030	. . . . . 6 ⊢ ∩ 𝑥 ∈ ∅ ∅ = V
5	4	eqeq1i 2770	. . . . 5 ⊢ (∩ 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅)
6	3, 5	mtbir 326	. . . 4 ⊢ ¬ ∩ 𝑥 ∈ ∅ ∅ = ∅
7		iineq1 4976	. . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∩ 𝑥 ∈ ∅ ∅)
8	7	eqeq1d 2767	. . . 4 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 ∅ = ∅ ↔ ∩ 𝑥 ∈ ∅ ∅ = ∅))
9	6, 8	mtbiri 330	. . 3 ⊢ (𝐴 = ∅ → ¬ ∩ 𝑥 ∈ 𝐴 ∅ = ∅)
10	9	necon2ai 2989	. 2 ⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅)
11	1, 10	impbii 212	1 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝑥 ∈ 𝐴 ∅ = ∅)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ≠ wne 2960  Vcvv 3457  ∅c0 4286  ∩ ciin 4959

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-nul 5271

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-v 3459  df-dif 3909  df-nul 4287  df-iin 4961

This theorem is used by: (None)