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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 13408 | . 2 ⊢ BOUNDED 𝑥 ∈ 𝐴 |
3 | vex 2715 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | snss 3685 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴) |
5 | 2, 4 | bd0 13386 | 1 ⊢ BOUNDED {𝑥} ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2128 ⊆ wss 3102 {csn 3560 BOUNDED wbd 13374 BOUNDED wbdc 13402 |
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-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-bd0 13375 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-in 3108 df-ss 3115 df-sn 3566 df-bdc 13403 |
This theorem is referenced by: bdvsn 13436 bdeqsuc 13443 |
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