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Mirrors > Home > ILE Home > Th. List > iin0r | GIF version |
Description: If an indexed intersection of the empty set is empty, the index set is nonempty. (Contributed by Jim Kingdon, 29-Aug-2018.) |
Ref | Expression |
---|---|
iin0r | ⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4055 | . . . . 5 ⊢ ∅ ∈ V | |
2 | n0i 3368 | . . . . 5 ⊢ (∅ ∈ V → ¬ V = ∅) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ ¬ V = ∅ |
4 | 0iin 3871 | . . . . 5 ⊢ ∩ 𝑥 ∈ ∅ ∅ = V | |
5 | 4 | eqeq1i 2147 | . . . 4 ⊢ (∩ 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅) |
6 | 3, 5 | mtbir 660 | . . 3 ⊢ ¬ ∩ 𝑥 ∈ ∅ ∅ = ∅ |
7 | iineq1 3827 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∩ 𝑥 ∈ ∅ ∅) | |
8 | 7 | eqeq1d 2148 | . . 3 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 ∅ = ∅ ↔ ∩ 𝑥 ∈ ∅ ∅ = ∅)) |
9 | 6, 8 | mtbiri 664 | . 2 ⊢ (𝐴 = ∅ → ¬ ∩ 𝑥 ∈ 𝐴 ∅ = ∅) |
10 | 9 | necon2ai 2362 | 1 ⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 Vcvv 2686 ∅c0 3363 ∩ ciin 3814 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-nul 4054 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-v 2688 df-dif 3073 df-nul 3364 df-iin 3816 |
This theorem is referenced by: (None) |
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