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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdcint | GIF version |
Description: The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
Ref | Expression |
---|---|
bdcint | ⊢ BOUNDED ∩ 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-bdel 13029 | . . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝑧 | |
2 | 1 | ax-bdal 13026 | . . . 4 ⊢ BOUNDED ∀𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 |
3 | df-ral 2421 | . . . 4 ⊢ (∀𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)) | |
4 | 2, 3 | bd0 13032 | . . 3 ⊢ BOUNDED ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧) |
5 | 4 | bdcab 13057 | . 2 ⊢ BOUNDED {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)} |
6 | df-int 3772 | . 2 ⊢ ∩ 𝑥 = {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)} | |
7 | 5, 6 | bdceqir 13052 | 1 ⊢ BOUNDED ∩ 𝑥 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 {cab 2125 ∀wral 2416 ∩ cint 3771 BOUNDED wbdc 13048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 ax-ext 2121 ax-bd0 13021 ax-bdal 13026 ax-bdel 13029 ax-bdsb 13030 |
This theorem depends on definitions: df-bi 116 df-clab 2126 df-cleq 2132 df-clel 2135 df-ral 2421 df-int 3772 df-bdc 13049 |
This theorem is referenced by: (None) |
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