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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 10612 | . . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝑧 | |
2 | 1 | ax-bdal 10609 | . . . 4 ⊢ BOUNDED ∀𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 |
3 | df-ral 2353 | . . . 4 ⊢ (∀𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)) | |
4 | 2, 3 | bd0 10615 | . . 3 ⊢ BOUNDED ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧) |
5 | 4 | bdcab 10640 | . 2 ⊢ BOUNDED {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)} |
6 | df-int 3637 | . 2 ⊢ ∩ 𝑥 = {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)} | |
7 | 5, 6 | bdceqir 10635 | 1 ⊢ BOUNDED ∩ 𝑥 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1282 {cab 2067 ∀wral 2348 ∩ cint 3636 BOUNDED wbdc 10631 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-17 1459 ax-ial 1467 ax-ext 2063 ax-bd0 10604 ax-bdal 10609 ax-bdel 10612 ax-bdsb 10613 |
This theorem depends on definitions: df-bi 115 df-clab 2068 df-cleq 2074 df-clel 2077 df-ral 2353 df-int 3637 df-bdc 10632 |
This theorem is referenced by: (None) |
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