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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 4114 | . . . . 5 ⊢ ∅ ∈ V | |
2 | n0i 3419 | . . . . 5 ⊢ (∅ ∈ V → ¬ V = ∅) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ ¬ V = ∅ |
4 | 0iin 3929 | . . . . 5 ⊢ ∩ 𝑥 ∈ ∅ ∅ = V | |
5 | 4 | eqeq1i 2178 | . . . 4 ⊢ (∩ 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅) |
6 | 3, 5 | mtbir 666 | . . 3 ⊢ ¬ ∩ 𝑥 ∈ ∅ ∅ = ∅ |
7 | iineq1 3885 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∩ 𝑥 ∈ ∅ ∅) | |
8 | 7 | eqeq1d 2179 | . . 3 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 ∅ = ∅ ↔ ∩ 𝑥 ∈ ∅ ∅ = ∅)) |
9 | 6, 8 | mtbiri 670 | . 2 ⊢ (𝐴 = ∅ → ¬ ∩ 𝑥 ∈ 𝐴 ∅ = ∅) |
10 | 9 | necon2ai 2394 | 1 ⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 Vcvv 2730 ∅c0 3414 ∩ ciin 3872 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-nul 4113 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-v 2732 df-dif 3123 df-nul 3415 df-iin 3874 |
This theorem is referenced by: (None) |
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