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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 13881 | . 2 ⊢ BOUNDED 𝑥 ∈ 𝐴 |
3 | vex 2733 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | snss 3709 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴) |
5 | 2, 4 | bd0 13859 | 1 ⊢ BOUNDED {𝑥} ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 ⊆ wss 3121 {csn 3583 BOUNDED wbd 13847 BOUNDED wbdc 13875 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-bd0 13848 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 df-ss 3134 df-sn 3589 df-bdc 13876 |
This theorem is referenced by: bdvsn 13909 bdeqsuc 13916 |
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