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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 11716 | . . . 4 ⊢ BOUNDED {𝑦} | |
2 | 1 | bdss 11710 | . . 3 ⊢ BOUNDED 𝑥 ⊆ {𝑦} |
3 | bdcv 11694 | . . . 4 ⊢ BOUNDED 𝑥 | |
4 | 3 | bdsnss 11719 | . . 3 ⊢ BOUNDED {𝑦} ⊆ 𝑥 |
5 | 2, 4 | ax-bdan 11661 | . 2 ⊢ BOUNDED (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥) |
6 | eqss 3040 | . 2 ⊢ (𝑥 = {𝑦} ↔ (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥)) | |
7 | 5, 6 | bd0r 11671 | 1 ⊢ BOUNDED 𝑥 = {𝑦} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1289 ⊆ wss 2999 {csn 3446 BOUNDED wbd 11658 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-bd0 11659 ax-bdan 11661 ax-bdal 11664 ax-bdeq 11666 ax-bdel 11667 ax-bdsb 11668 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-v 2621 df-in 3005 df-ss 3012 df-sn 3452 df-bdc 11687 |
This theorem is referenced by: bdop 11721 |
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