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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 9325 | . . . 4 ⊢ BOUNDED {y} | |
2 | 1 | bdss 9319 | . . 3 ⊢ BOUNDED x ⊆ {y} |
3 | bdcv 9303 | . . . 4 ⊢ BOUNDED x | |
4 | 3 | bdsnss 9328 | . . 3 ⊢ BOUNDED {y} ⊆ x |
5 | 2, 4 | ax-bdan 9270 | . 2 ⊢ BOUNDED (x ⊆ {y} ∧ {y} ⊆ x) |
6 | eqss 2954 | . 2 ⊢ (x = {y} ↔ (x ⊆ {y} ∧ {y} ⊆ x)) | |
7 | 5, 6 | bd0r 9280 | 1 ⊢ BOUNDED x = {y} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1242 ⊆ wss 2911 {csn 3367 BOUNDED wbd 9267 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-bd0 9268 ax-bdan 9270 ax-bdal 9273 ax-bdeq 9275 ax-bdel 9276 ax-bdsb 9277 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-ral 2305 df-v 2553 df-in 2918 df-ss 2925 df-sn 3373 df-bdc 9296 |
This theorem is referenced by: bdop 9330 |
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