Theorem iin0 5291

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 4932	. 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∅)
2		0ex 5229	. . . . . 6 ⊢ ∅ ∈ V
3	2	n0ii 4271	. . . . 5 ⊢ ¬ V = ∅
4		0iin 4993	. . . . . 6 ⊢ ∩ 𝑥 ∈ ∅ ∅ = V
5	4	eqeq1i 2744	. . . . 5 ⊢ (∩ 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅)
6	3, 5	mtbir 324	. . . 4 ⊢ ¬ ∩ 𝑥 ∈ ∅ ∅ = ∅
7		iineq1 4939	. . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∩ 𝑥 ∈ ∅ ∅)
8	7	eqeq1d 2741	. . . 4 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 ∅ = ∅ ↔ ∩ 𝑥 ∈ ∅ ∅ = ∅))
9	6, 8	mtbiri 328	. . 3 ⊢ (𝐴 = ∅ → ¬ ∩ 𝑥 ∈ 𝐴 ∅ = ∅)
10	9	necon2ai 2963	. 2 ⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅)
11	1, 10	impbii 210	1 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝑥 ∈ 𝐴 ∅ = ∅)

Syntax hints:   ↔ wb 207   = wceq 1547   ≠ wne 2934  Vcvv 3431  ∅c0 4261  ∩ ciin 4922

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-v 3433  df-dif 3886  df-nul 4262  df-iin 4924

This theorem is referenced by: (None)