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Mirrors > Home > MPE Home > Th. List > iin0 | Structured version Visualization version GIF version |
Description: An indexed intersection of the empty set, with a nonempty index set, is empty. (Contributed by NM, 20-Oct-2005.) |
Ref | Expression |
---|---|
iin0 | ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝑥 ∈ 𝐴 ∅ = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iinconst 5008 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∅) | |
2 | 0ex 5308 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | 2 | n0ii 4337 | . . . . 5 ⊢ ¬ V = ∅ |
4 | 0iin 5068 | . . . . . 6 ⊢ ∩ 𝑥 ∈ ∅ ∅ = V | |
5 | 4 | eqeq1i 2738 | . . . . 5 ⊢ (∩ 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅) |
6 | 3, 5 | mtbir 323 | . . . 4 ⊢ ¬ ∩ 𝑥 ∈ ∅ ∅ = ∅ |
7 | iineq1 5015 | . . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 ∅ = ∩ 𝑥 ∈ ∅ ∅) | |
8 | 7 | eqeq1d 2735 | . . . 4 ⊢ (𝐴 = ∅ → (∩ 𝑥 ∈ 𝐴 ∅ = ∅ ↔ ∩ 𝑥 ∈ ∅ ∅ = ∅)) |
9 | 6, 8 | mtbiri 327 | . . 3 ⊢ (𝐴 = ∅ → ¬ ∩ 𝑥 ∈ 𝐴 ∅ = ∅) |
10 | 9 | necon2ai 2971 | . 2 ⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅) |
11 | 1, 10 | impbii 208 | 1 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝑥 ∈ 𝐴 ∅ = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ≠ wne 2941 Vcvv 3475 ∅c0 4323 ∩ ciin 4999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-12 2172 ax-ext 2704 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-v 3477 df-dif 3952 df-nul 4324 df-iin 5001 |
This theorem is referenced by: (None) |
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