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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 13463 | . . 3 ⊢ BOUNDED 𝑦 ∈ 𝐴 |
3 | 2 | ax-bdal 13435 | . 2 ⊢ BOUNDED ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 |
4 | dfss3 3118 | . 2 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴) | |
5 | 3, 4 | bd0r 13442 | 1 ⊢ BOUNDED 𝑥 ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2128 ∀wral 2435 ⊆ wss 3102 BOUNDED wbd 13429 BOUNDED wbdc 13457 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-11 1486 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-bd0 13430 ax-bdal 13435 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-ral 2440 df-in 3108 df-ss 3115 df-bdc 13458 |
This theorem is referenced by: bdeq0 13484 bdcpw 13486 bdvsn 13491 bdop 13492 bdeqsuc 13498 bj-nntrans 13568 bj-omtrans 13573 |
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