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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 15036 | . 2 ⊢ BOUNDED 𝑥 ∈ 𝐴 |
3 | vex 2755 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | snss 3742 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴) |
5 | 2, 4 | bd0 15014 | 1 ⊢ BOUNDED {𝑥} ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 ⊆ wss 3144 {csn 3607 BOUNDED wbd 15002 BOUNDED wbdc 15030 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-bd0 15003 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-ss 3157 df-sn 3613 df-bdc 15031 |
This theorem is referenced by: bdvsn 15064 bdeqsuc 15071 |
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