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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdvsn | GIF version |
Description: Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
Ref | Expression |
---|---|
bdvsn | ⊢ BOUNDED 𝑥 = {𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdcsn 13057 | . . . 4 ⊢ BOUNDED {𝑦} | |
2 | 1 | bdss 13051 | . . 3 ⊢ BOUNDED 𝑥 ⊆ {𝑦} |
3 | bdcv 13035 | . . . 4 ⊢ BOUNDED 𝑥 | |
4 | 3 | bdsnss 13060 | . . 3 ⊢ BOUNDED {𝑦} ⊆ 𝑥 |
5 | 2, 4 | ax-bdan 13002 | . 2 ⊢ BOUNDED (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥) |
6 | eqss 3107 | . 2 ⊢ (𝑥 = {𝑦} ↔ (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥)) | |
7 | 5, 6 | bd0r 13012 | 1 ⊢ BOUNDED 𝑥 = {𝑦} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ⊆ wss 3066 {csn 3522 BOUNDED wbd 12999 |
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 ax-bdan 13002 ax-bdal 13005 ax-bdeq 13007 ax-bdel 13008 ax-bdsb 13009 |
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-ral 2419 df-v 2683 df-in 3072 df-ss 3079 df-sn 3528 df-bdc 13028 |
This theorem is referenced by: bdop 13062 |
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