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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 16462 | . 2 ⊢ BOUNDED 𝑥 ∈ 𝐴 |
| 3 | vex 2805 | . . 3 ⊢ 𝑥 ∈ V | |
| 4 | 3 | snss 3808 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴) |
| 5 | 2, 4 | bd0 16440 | 1 ⊢ BOUNDED {𝑥} ⊆ 𝐴 |
| Colors of variables: wff set class |
| Syntax hints: ∈ wcel 2202 ⊆ wss 3200 {csn 3669 BOUNDED wbd 16428 BOUNDED wbdc 16456 |
| 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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-ext 2213 ax-bd0 16429 |
| This theorem depends on definitions: df-bi 117 df-tru 1400 df-nf 1509 df-sb 1811 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-v 2804 df-in 3206 df-ss 3213 df-sn 3675 df-bdc 16457 |
| This theorem is referenced by: bdvsn 16490 bdeqsuc 16497 |
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