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Mirrors > Home > MPE Home > Th. List > Mathboxes > fipjust | Structured version Visualization version GIF version |
Description: A definition of the finite intersection property of a class based on closure under pair-wise intersection of its elements is independent of the dummy variables. (Contributed by Richard Penner, 1-Jan-2020.) |
Ref | Expression |
---|---|
fipjust | ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 ∩ 𝑣) ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1 3769 | . . 3 ⊢ (𝑢 = 𝑥 → (𝑢 ∩ 𝑣) = (𝑥 ∩ 𝑣)) | |
2 | 1 | eleq1d 2672 | . 2 ⊢ (𝑢 = 𝑥 → ((𝑢 ∩ 𝑣) ∈ 𝐴 ↔ (𝑥 ∩ 𝑣) ∈ 𝐴)) |
3 | ineq2 3770 | . . 3 ⊢ (𝑣 = 𝑦 → (𝑥 ∩ 𝑣) = (𝑥 ∩ 𝑦)) | |
4 | 3 | eleq1d 2672 | . 2 ⊢ (𝑣 = 𝑦 → ((𝑥 ∩ 𝑣) ∈ 𝐴 ↔ (𝑥 ∩ 𝑦) ∈ 𝐴)) |
5 | 2, 4 | cbvral2v 3155 | 1 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 ∩ 𝑣) ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∈ wcel 1977 ∀wral 2896 ∩ cin 3539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-in 3547 |
This theorem is referenced by: (None) |
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