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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 2576 | . 2 ⊢ (∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → ∃𝑦〈𝐴, 𝑦〉 ∈ 𝑉) | |
| 2 | nfeu1 2587 | . . . 4 ⊢ Ⅎ𝑦∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 | |
| 3 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
| 4 | 3 | nfel1 2921 | . . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V | 
| 5 | 2, 4 | nfim 1895 | . . 3 ⊢ Ⅎ𝑦(∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → 𝐴 ∈ V) | 
| 6 | opprc1 4896 | . . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V → 〈𝐴, 𝑦〉 = ∅) | |
| 7 | 6 | eleq1d 2825 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (〈𝐴, 𝑦〉 ∈ 𝑉 ↔ ∅ ∈ 𝑉)) | 
| 8 | ax-5 1909 | . . . . . . . . 9 ⊢ (∅ ∈ 𝑉 → ∀𝑦∅ ∈ 𝑉) | |
| 9 | alneu 47141 | . . . . . . . . 9 ⊢ (∀𝑦∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉) | |
| 10 | 8, 9 | syl 17 | . . . . . . . 8 ⊢ (∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉) | 
| 11 | 7, 10 | biimtrdi 253 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (〈𝐴, 𝑦〉 ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)) | 
| 12 | 11 | impcom 407 | . . . . . 6 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦∅ ∈ 𝑉) | 
| 13 | 7 | eubidv 2585 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 ↔ ∃!𝑦∅ ∈ 𝑉)) | 
| 14 | 13 | notbid 318 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (¬ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉)) | 
| 15 | 14 | adantl 481 | . . . . . 6 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → (¬ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉)) | 
| 16 | 12, 15 | mpbird 257 | . . . . 5 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉) | 
| 17 | 16 | ex 412 | . . . 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 2216 | . 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 206 ∧ wa 395 ∀wal 1537 ∃wex 1778 ∈ wcel 2107 ∃!weu 2567 Vcvv 3479 ∅c0 4332 〈cop 4631 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-nul 5305 ax-pow 5364 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-dif 3953 df-ss 3967 df-nul 4333 df-if 4525 df-op 4632 | 
| This theorem is referenced by: afveu 47170 tz6.12-afv 47190 tz6.12-afv2 47257 | 
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