| Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
| 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 3484 | . . 3 ⊢ 𝑥 ∈ V | |
| 3 | 2 | inex1 5317 | . 2 ⊢ (𝑥 ∩ 𝑦) ∈ V |
| 4 | sseq2 4010 | . 2 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ (𝑥 ∩ 𝑦))) | |
| 5 | sseq2 4010 | . 2 ⊢ (𝑧 = 𝑥 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑥)) | |
| 6 | sseq2 4010 | . 2 ⊢ (𝑧 = 𝑦 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑦)) | |
| 7 | ssin 4239 | . . 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 43579 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-in 3958 df-ss 3968 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |