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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 13033 | . 2 ⊢ BOUNDED 𝑥 ∈ 𝐴 |
3 | vex 2684 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | snss 3644 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴) |
5 | 2, 4 | bd0 13011 | 1 ⊢ BOUNDED {𝑥} ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 ⊆ wss 3066 {csn 3522 BOUNDED wbd 12999 BOUNDED wbdc 13027 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-bd0 13000 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-ss 3079 df-sn 3528 df-bdc 13028 |
This theorem is referenced by: bdvsn 13061 bdeqsuc 13068 |
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