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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssficl | Structured version Visualization version GIF version |
Description: The class of all subsets of a class has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 3-Jan-2020.) |
Ref | Expression |
---|---|
ssficl.a | ⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵} |
Ref | Expression |
---|---|
ssficl | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssficl.a | . 2 ⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵} | |
2 | vex 3475 | . . 3 ⊢ 𝑥 ∈ V | |
3 | 2 | inex1 5317 | . 2 ⊢ (𝑥 ∩ 𝑦) ∈ V |
4 | sseq1 4005 | . 2 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ⊆ 𝐵 ↔ (𝑥 ∩ 𝑦) ⊆ 𝐵)) | |
5 | sseq1 4005 | . 2 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵)) | |
6 | sseq1 4005 | . 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵)) | |
7 | ssinss1 4238 | . . 3 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∩ 𝑦) ⊆ 𝐵) | |
8 | 7 | adantr 480 | . 2 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → (𝑥 ∩ 𝑦) ⊆ 𝐵) |
9 | 1, 3, 4, 5, 6, 8 | cllem0 42996 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 {cab 2705 ∀wral 3058 Vcvv 3471 ∩ cin 3946 ⊆ wss 3947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-v 3473 df-in 3954 df-ss 3964 |
This theorem is referenced by: (None) |
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