Theorem iin0 5360

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 5007	. 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∅)
2		0ex 5307	. . . . . 6 ⊢ ∅ ∈ V
3	2	n0ii 4336	. . . . 5 ⊢ ¬ V = ∅
4		0iin 5067	. . . . . 6 ⊢ ∩ 𝑥 ∈ ∅ ∅ = V
5	4	eqeq1i 2737	. . . . 5 ⊢ (∩ 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅)
6	3, 5	mtbir 322	. . . 4 ⊢ ¬ ∩ 𝑥 ∈ ∅ ∅ = ∅
7		iineq1 5014	. . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∩ 𝑥 ∈ ∅ ∅)
8	7	eqeq1d 2734	. . . 4 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 ∅ = ∅ ↔ ∩ 𝑥 ∈ ∅ ∅ = ∅))
9	6, 8	mtbiri 326	. . 3 ⊢ (𝐴 = ∅ → ¬ ∩ 𝑥 ∈ 𝐴 ∅ = ∅)
10	9	necon2ai 2970	. 2 ⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅)
11	1, 10	impbii 208	1 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝑥 ∈ 𝐴 ∅ = ∅)

Syntax hints:   ↔ wb 205   = wceq 1541   ≠ wne 2940  Vcvv 3474  ∅c0 4322  ∩ ciin 4998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3476  df-dif 3951  df-nul 4323  df-iin 5000

This theorem is referenced by: (None)