Theorem iin0r 4148

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 4109	. . . . 5 ⊢ ∅ ∈ V
2		n0i 3414	. . . . 5 ⊢ (∅ ∈ V → ¬ V = ∅)
3	1, 2	ax-mp 5	. . . 4 ⊢ ¬ V = ∅
4		0iin 3924	. . . . 5 ⊢ ∩ 𝑥 ∈ ∅ ∅ = V
5	4	eqeq1i 2173	. . . 4 ⊢ (∩ 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅)
6	3, 5	mtbir 661	. . 3 ⊢ ¬ ∩ 𝑥 ∈ ∅ ∅ = ∅
7		iineq1 3880	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∩ 𝑥 ∈ ∅ ∅)
8	7	eqeq1d 2174	. . 3 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 ∅ = ∅ ↔ ∩ 𝑥 ∈ ∅ ∅ = ∅))
9	6, 8	mtbiri 665	. 2 ⊢ (𝐴 = ∅ → ¬ ∩ 𝑥 ∈ 𝐴 ∅ = ∅)
10	9	necon2ai 2390	1 ⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1343   ∈ wcel 2136   ≠ wne 2336  Vcvv 2726  ∅c0 3409  ∩ ciin 3867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-nul 4108

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-v 2728  df-dif 3118  df-nul 3410  df-iin 3869

This theorem is referenced by: (None)