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Mirrors > Home > MPE Home > Th. List > Mathboxes > superficl | Structured version Visualization version GIF version |
Description: The class of all supersets of a class has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 3-Jan-2020.) |
Ref | Expression |
---|---|
superficl.a | ⊢ 𝐴 = {𝑧 ∣ 𝐵 ⊆ 𝑧} |
Ref | Expression |
---|---|
superficl | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | superficl.a | . 2 ⊢ 𝐴 = {𝑧 ∣ 𝐵 ⊆ 𝑧} | |
2 | vex 3481 | . . 3 ⊢ 𝑥 ∈ V | |
3 | 2 | inex1 5322 | . 2 ⊢ (𝑥 ∩ 𝑦) ∈ V |
4 | sseq2 4021 | . 2 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ (𝑥 ∩ 𝑦))) | |
5 | sseq2 4021 | . 2 ⊢ (𝑧 = 𝑥 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑥)) | |
6 | sseq2 4021 | . 2 ⊢ (𝑧 = 𝑦 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑦)) | |
7 | ssin 4246 | . . 3 ⊢ ((𝐵 ⊆ 𝑥 ∧ 𝐵 ⊆ 𝑦) ↔ 𝐵 ⊆ (𝑥 ∩ 𝑦)) | |
8 | 7 | biimpi 216 | . 2 ⊢ ((𝐵 ⊆ 𝑥 ∧ 𝐵 ⊆ 𝑦) → 𝐵 ⊆ (𝑥 ∩ 𝑦)) |
9 | 1, 3, 4, 5, 6, 8 | cllem0 43555 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1536 ∈ wcel 2105 {cab 2711 ∀wral 3058 Vcvv 3477 ∩ cin 3961 ⊆ wss 3962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-v 3479 df-in 3969 df-ss 3979 |
This theorem is referenced by: (None) |
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