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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 4103 | . . . . 5 ⊢ ∅ ∈ V | |
2 | n0i 3409 | . . . . 5 ⊢ (∅ ∈ V → ¬ V = ∅) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ ¬ V = ∅ |
4 | 0iin 3918 | . . . . 5 ⊢ ∩ 𝑥 ∈ ∅ ∅ = V | |
5 | 4 | eqeq1i 2172 | . . . 4 ⊢ (∩ 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅) |
6 | 3, 5 | mtbir 661 | . . 3 ⊢ ¬ ∩ 𝑥 ∈ ∅ ∅ = ∅ |
7 | iineq1 3874 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∩ 𝑥 ∈ ∅ ∅) | |
8 | 7 | eqeq1d 2173 | . . 3 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 ∅ = ∅ ↔ ∩ 𝑥 ∈ ∅ ∅ = ∅)) |
9 | 6, 8 | mtbiri 665 | . 2 ⊢ (𝐴 = ∅ → ¬ ∩ 𝑥 ∈ 𝐴 ∅ = ∅) |
10 | 9 | necon2ai 2388 | 1 ⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1342 ∈ wcel 2135 ≠ wne 2334 Vcvv 2721 ∅c0 3404 ∩ ciin 3861 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-nul 4102 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-v 2723 df-dif 3113 df-nul 3405 df-iin 3863 |
This theorem is referenced by: (None) |
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