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Mirrors > Home > NFE Home > Th. List > rabsneu | GIF version |
Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
rabsneu | ⊢ ((A ∈ V ∧ {x ∈ B ∣ φ} = {A}) → ∃!x ∈ B φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 2623 | . . . 4 ⊢ {x ∈ B ∣ φ} = {x ∣ (x ∈ B ∧ φ)} | |
2 | 1 | eqeq1i 2360 | . . 3 ⊢ ({x ∈ B ∣ φ} = {A} ↔ {x ∣ (x ∈ B ∧ φ)} = {A}) |
3 | absneu 3794 | . . 3 ⊢ ((A ∈ V ∧ {x ∣ (x ∈ B ∧ φ)} = {A}) → ∃!x(x ∈ B ∧ φ)) | |
4 | 2, 3 | sylan2b 461 | . 2 ⊢ ((A ∈ V ∧ {x ∈ B ∣ φ} = {A}) → ∃!x(x ∈ B ∧ φ)) |
5 | df-reu 2621 | . 2 ⊢ (∃!x ∈ B φ ↔ ∃!x(x ∈ B ∧ φ)) | |
6 | 4, 5 | sylibr 203 | 1 ⊢ ((A ∈ V ∧ {x ∈ B ∣ φ} = {A}) → ∃!x ∈ B φ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∃!weu 2204 {cab 2339 ∃!wreu 2616 {crab 2618 {csn 3737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-reu 2621 df-rab 2623 df-v 2861 df-sn 3741 |
This theorem is referenced by: (None) |
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