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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 3499 | . . 3 ⊢ 𝑥 ∈ V | |
3 | 2 | inex1 5223 | . 2 ⊢ (𝑥 ∩ 𝑦) ∈ V |
4 | sseq2 3995 | . 2 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ (𝑥 ∩ 𝑦))) | |
5 | sseq2 3995 | . 2 ⊢ (𝑧 = 𝑥 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑥)) | |
6 | sseq2 3995 | . 2 ⊢ (𝑧 = 𝑦 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑦)) | |
7 | ssin 4209 | . . 3 ⊢ ((𝐵 ⊆ 𝑥 ∧ 𝐵 ⊆ 𝑦) ↔ 𝐵 ⊆ (𝑥 ∩ 𝑦)) | |
8 | 7 | biimpi 218 | . 2 ⊢ ((𝐵 ⊆ 𝑥 ∧ 𝐵 ⊆ 𝑦) → 𝐵 ⊆ (𝑥 ∩ 𝑦)) |
9 | 1, 3, 4, 5, 6, 8 | cllem0 39932 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2801 ∀wral 3140 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-v 3498 df-in 3945 df-ss 3954 |
This theorem is referenced by: (None) |
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