Theorem iin0r 4281

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 4236	. . . . 5 ⊢ ∅ ∈ V
2		n0i 3513	. . . . 5 ⊢ (∅ ∈ V → ¬ V = ∅)
3	1, 2	ax-mp 5	. . . 4 ⊢ ¬ V = ∅
4		0iin 4049	. . . . 5 ⊢ ∩ 𝑥 ∈ ∅ ∅ = V
5	4	eqeq1i 2240	. . . 4 ⊢ (∩ 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅)
6	3, 5	mtbir 678	. . 3 ⊢ ¬ ∩ 𝑥 ∈ ∅ ∅ = ∅
7		iineq1 4004	. . . 4 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∩ 𝑥 ∈ ∅ ∅)
8	7	eqeq1d 2241	. . 3 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 ∅ = ∅ ↔ ∩ 𝑥 ∈ ∅ ∅ = ∅))
9	6, 8	mtbiri 682	. 2 ⊢ (𝐴 = ∅ → ¬ ∩ 𝑥 ∈ 𝐴 ∅ = ∅)
10	9	necon2ai 2466	1 ⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1398   ∈ wcel 2203   ≠ wne 2412  Vcvv 2812  ∅c0 3507  ∩ ciin 3991

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-nul 4235

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-v 2814  df-dif 3212  df-nul 3508  df-iin 3993

This theorem is referenced by: (None)