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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdsnss | GIF version |
Description: Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
Ref | Expression |
---|---|
bdsnss.1 | ⊢ BOUNDED 𝐴 |
Ref | Expression |
---|---|
bdsnss | ⊢ BOUNDED {𝑥} ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdsnss.1 | . . 3 ⊢ BOUNDED 𝐴 | |
2 | 1 | bdeli 15459 | . 2 ⊢ BOUNDED 𝑥 ∈ 𝐴 |
3 | vex 2766 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | snss 3757 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴) |
5 | 2, 4 | bd0 15437 | 1 ⊢ BOUNDED {𝑥} ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2167 ⊆ wss 3157 {csn 3622 BOUNDED wbd 15425 BOUNDED wbdc 15453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-ext 2178 ax-bd0 15426 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1475 df-sb 1777 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-v 2765 df-in 3163 df-ss 3170 df-sn 3628 df-bdc 15454 |
This theorem is referenced by: bdvsn 15487 bdeqsuc 15494 |
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