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Mirrors > Home > MPE Home > Th. List > Mathboxes > eu2ndop1stv | Structured version Visualization version GIF version |
Description: If there is a unique second component in an ordered pair contained in a given set, the first component must be a set. (Contributed by Alexander van der Vekens, 29-Nov-2017.) |
Ref | Expression |
---|---|
eu2ndop1stv | ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euex 2570 | . 2 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉) | |
2 | nfeu1 2581 | . . . 4 ⊢ Ⅎ𝑦∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 | |
3 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
4 | 3 | nfel1 2918 | . . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V |
5 | 2, 4 | nfim 1899 | . . 3 ⊢ Ⅎ𝑦(∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V) |
6 | opprc1 4874 | . . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝑦⟩ = ∅) | |
7 | 6 | eleq1d 2817 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ∅ ∈ 𝑉)) |
8 | ax-5 1913 | . . . . . . . . 9 ⊢ (∅ ∈ 𝑉 → ∀𝑦∅ ∈ 𝑉) | |
9 | alneu 45509 | . . . . . . . . 9 ⊢ (∀𝑦∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉) | |
10 | 8, 9 | syl 17 | . . . . . . . 8 ⊢ (∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉) |
11 | 7, 10 | syl6bi 252 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)) |
12 | 11 | impcom 408 | . . . . . 6 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦∅ ∈ 𝑉) |
13 | 7 | eubidv 2579 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ∃!𝑦∅ ∈ 𝑉)) |
14 | 13 | notbid 317 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (¬ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉)) |
15 | 14 | adantl 482 | . . . . . 6 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → (¬ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉)) |
16 | 12, 15 | mpbird 256 | . . . . 5 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉) |
17 | 16 | ex 413 | . . . 4 ⊢ (⟨𝐴, 𝑦⟩ ∈ 𝑉 → (¬ 𝐴 ∈ V → ¬ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉)) |
18 | 17 | con4d 115 | . . 3 ⊢ (⟨𝐴, 𝑦⟩ ∈ 𝑉 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V)) |
19 | 5, 18 | exlimi 2210 | . 2 ⊢ (∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V)) |
20 | 1, 19 | mpcom 38 | 1 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 ∃wex 1781 ∈ wcel 2106 ∃!weu 2561 Vcvv 3459 ∅c0 4302 ⟨cop 4612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-nul 5283 ax-pow 5340 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-v 3461 df-dif 3931 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-op 4613 |
This theorem is referenced by: afveu 45538 tz6.12-afv 45558 tz6.12-afv2 45625 |
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