Theorem iin0r 4198

Description: If an indexed intersection of the empty set is empty, the index set is nonempty. (Contributed by Jim Kingdon, 29-Aug-2018.)

Assertion

Ref	Expression
iin0r	⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅)

Distinct variable group:   𝑥,𝐴

Proof of Theorem iin0r

Step	Hyp	Ref	Expression
1		0ex 4156	. . . . 5 ⊢ ∅ ∈ V
2		n0i 3452	. . . . 5 ⊢ (∅ ∈ V → ¬ V = ∅)
3	1, 2	ax-mp 5	. . . 4 ⊢ ¬ V = ∅
4		0iin 3971	. . . . 5 ⊢ ∩ 𝑥 ∈ ∅ ∅ = V
5	4	eqeq1i 2201	. . . 4 ⊢ (∩ 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅)
6	3, 5	mtbir 672	. . 3 ⊢ ¬ ∩ 𝑥 ∈ ∅ ∅ = ∅
7		iineq1 3926	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∩ 𝑥 ∈ ∅ ∅)
8	7	eqeq1d 2202	. . 3 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 ∅ = ∅ ↔ ∩ 𝑥 ∈ ∅ ∅ = ∅))
9	6, 8	mtbiri 676	. 2 ⊢ (𝐴 = ∅ → ¬ ∩ 𝑥 ∈ 𝐴 ∅ = ∅)
10	9	necon2ai 2418	1 ⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1364   ∈ wcel 2164   ≠ wne 2364  Vcvv 2760  ∅c0 3446  ∩ ciin 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-nul 4155

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-v 2762  df-dif 3155  df-nul 3447  df-iin 3915

This theorem is referenced by: (None)