| Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
| 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 3484 | . . 3 ⊢ 𝑥 ∈ V | |
| 3 | 2 | inex1 5317 | . 2 ⊢ (𝑥 ∩ 𝑦) ∈ V |
| 4 | sseq1 4009 | . 2 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ⊆ 𝐵 ↔ (𝑥 ∩ 𝑦) ⊆ 𝐵)) | |
| 5 | sseq1 4009 | . 2 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵)) | |
| 6 | sseq1 4009 | . 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵)) | |
| 7 | ssinss1 4246 | . . 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 43579 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = 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 |