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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdss | GIF version |
Description: The inclusion of a setvar in a bounded class is a bounded formula. Note: apparently, we cannot prove from the present axioms that equality of two bounded classes is a bounded formula. (Contributed by BJ, 3-Oct-2019.) |
Ref | Expression |
---|---|
bdss.1 | ⊢ BOUNDED 𝐴 |
Ref | Expression |
---|---|
bdss | ⊢ BOUNDED 𝑥 ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdss.1 | . . . 4 ⊢ BOUNDED 𝐴 | |
2 | 1 | bdeli 12971 | . . 3 ⊢ BOUNDED 𝑦 ∈ 𝐴 |
3 | 2 | ax-bdal 12943 | . 2 ⊢ BOUNDED ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 |
4 | dfss3 3057 | . 2 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) | |
5 | 3, 4 | bd0r 12950 | 1 ⊢ BOUNDED 𝑥 ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 ∀wral 2393 ⊆ wss 3041 BOUNDED wbd 12937 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-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-bd0 12938 ax-bdal 12943 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-ral 2398 df-in 3047 df-ss 3054 df-bdc 12966 |
This theorem is referenced by: bdeq0 12992 bdcpw 12994 bdvsn 12999 bdop 13000 bdeqsuc 13006 bj-nntrans 13076 bj-omtrans 13081 |
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