Theorem iin0r 4259

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 4216	. . . . 5 ⊢ ∅ ∈ V
2		n0i 3500	. . . . 5 ⊢ (∅ ∈ V → ¬ V = ∅)
3	1, 2	ax-mp 5	. . . 4 ⊢ ¬ V = ∅
4		0iin 4029	. . . . 5 ⊢ ∩ 𝑥 ∈ ∅ ∅ = V
5	4	eqeq1i 2239	. . . 4 ⊢ (∩ 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅)
6	3, 5	mtbir 677	. . 3 ⊢ ¬ ∩ 𝑥 ∈ ∅ ∅ = ∅
7		iineq1 3984	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∩ 𝑥 ∈ ∅ ∅)
8	7	eqeq1d 2240	. . 3 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 ∅ = ∅ ↔ ∩ 𝑥 ∈ ∅ ∅ = ∅))
9	6, 8	mtbiri 681	. 2 ⊢ (𝐴 = ∅ → ¬ ∩ 𝑥 ∈ 𝐴 ∅ = ∅)
10	9	necon2ai 2456	1 ⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1397   ∈ wcel 2202   ≠ wne 2402  Vcvv 2802  ∅c0 3494  ∩ ciin 3971

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-nul 4215

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-v 2804  df-dif 3202  df-nul 3495  df-iin 3973

This theorem is referenced by: (None)