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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 12946 | . . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝑧 | |
2 | 1 | ax-bdal 12943 | . . . 4 ⊢ BOUNDED ∀𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 |
3 | df-ral 2398 | . . . 4 ⊢ (∀𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)) | |
4 | 2, 3 | bd0 12949 | . . 3 ⊢ BOUNDED ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧) |
5 | 4 | bdcab 12974 | . 2 ⊢ BOUNDED {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)} |
6 | df-int 3742 | . 2 ⊢ ∩ 𝑥 = {𝑦 ∣ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)} | |
7 | 5, 6 | bdceqir 12969 | 1 ⊢ BOUNDED ∩ 𝑥 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1314 {cab 2103 ∀wral 2393 ∩ cint 3741 BOUNDED wbdc 12965 |
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 1408 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-4 1472 ax-17 1491 ax-ial 1499 ax-ext 2099 ax-bd0 12938 ax-bdal 12943 ax-bdel 12946 ax-bdsb 12947 |
This theorem depends on definitions: df-bi 116 df-clab 2104 df-cleq 2110 df-clel 2113 df-ral 2398 df-int 3742 df-bdc 12966 |
This theorem is referenced by: (None) |
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