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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdrmo | GIF version |
Description: Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.) |
Ref | Expression |
---|---|
bdrmo.1 | ⊢ BOUNDED 𝜑 |
Ref | Expression |
---|---|
bdrmo | ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdrmo.1 | . . . 4 ⊢ BOUNDED 𝜑 | |
2 | 1 | ax-bdex 14227 | . . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑 |
3 | 1 | bdreu 14263 | . . 3 ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑 |
4 | 2, 3 | ax-bdim 14222 | . 2 ⊢ BOUNDED (∃𝑥 ∈ 𝑦 𝜑 → ∃!𝑥 ∈ 𝑦 𝜑) |
5 | rmo5 2692 | . 2 ⊢ (∃*𝑥 ∈ 𝑦 𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 → ∃!𝑥 ∈ 𝑦 𝜑)) | |
6 | 4, 5 | bd0r 14233 | 1 ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃wrex 2456 ∃!wreu 2457 ∃*wrmo 2458 BOUNDED wbd 14220 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-bd0 14221 ax-bdim 14222 ax-bdan 14223 ax-bdal 14226 ax-bdex 14227 ax-bdeq 14228 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-cleq 2170 df-clel 2173 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 |
This theorem is referenced by: (None) |
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