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Mirrors > Home > ILE Home > Th. List > iin0imm | GIF version |
Description: An indexed intersection of the empty set, with an inhabited index set, is empty. (Contributed by Jim Kingdon, 29-Aug-2018.) |
Ref | Expression |
---|---|
iin0imm | ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 ∅ = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iinconstm 3822 | 1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 ∅ = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∅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-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 |
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-ral 2421 df-v 2688 df-iin 3816 |
This theorem is referenced by: (None) |
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