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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 non-empty. (Contributed by Jim Kingdon, 29-Aug-2018.) |
Ref | Expression |
---|---|
iin0r | ⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 3884 | . . . . 5 ⊢ ∅ ∈ V | |
2 | n0i 3229 | . . . . 5 ⊢ (∅ ∈ V → ¬ V = ∅) | |
3 | 1, 2 | ax-mp 7 | . . . 4 ⊢ ¬ V = ∅ |
4 | 0iin 3715 | . . . . 5 ⊢ ∩ 𝑥 ∈ ∅ ∅ = V | |
5 | 4 | eqeq1i 2047 | . . . 4 ⊢ (∩ 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅) |
6 | 3, 5 | mtbir 596 | . . 3 ⊢ ¬ ∩ 𝑥 ∈ ∅ ∅ = ∅ |
7 | iineq1 3671 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∩ 𝑥 ∈ ∅ ∅) | |
8 | 7 | eqeq1d 2048 | . . 3 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 ∅ = ∅ ↔ ∩ 𝑥 ∈ ∅ ∅ = ∅)) |
9 | 6, 8 | mtbiri 600 | . 2 ⊢ (𝐴 = ∅ → ¬ ∩ 𝑥 ∈ 𝐴 ∅ = ∅) |
10 | 9 | necon2ai 2259 | 1 ⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1243 ∈ wcel 1393 ≠ wne 2204 Vcvv 2557 ∅c0 3224 ∩ ciin 3658 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-nul 3883 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-v 2559 df-dif 2920 df-nul 3225 df-iin 3660 |
This theorem is referenced by: (None) |
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