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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 15075 | . . . 4 ⊢ BOUNDED {𝑦} | |
2 | 1 | bdss 15069 | . . 3 ⊢ BOUNDED 𝑥 ⊆ {𝑦} |
3 | bdcv 15053 | . . . 4 ⊢ BOUNDED 𝑥 | |
4 | 3 | bdsnss 15078 | . . 3 ⊢ BOUNDED {𝑦} ⊆ 𝑥 |
5 | 2, 4 | ax-bdan 15020 | . 2 ⊢ BOUNDED (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥) |
6 | eqss 3185 | . 2 ⊢ (𝑥 = {𝑦} ↔ (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥)) | |
7 | 5, 6 | bd0r 15030 | 1 ⊢ BOUNDED 𝑥 = {𝑦} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1364 ⊆ wss 3144 {csn 3607 BOUNDED wbd 15017 |
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 15018 ax-bdan 15020 ax-bdal 15023 ax-bdeq 15025 ax-bdel 15026 ax-bdsb 15027 |
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-ral 2473 df-v 2754 df-in 3150 df-ss 3157 df-sn 3613 df-bdc 15046 |
This theorem is referenced by: bdop 15080 |
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