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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 2482 | . 2 ⊢ (∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → ∃𝑦〈𝐴, 𝑦〉 ∈ 𝑉) | |
2 | nfeu1 2468 | . . . 4 ⊢ Ⅎ𝑦∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 | |
3 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
4 | 3 | nfel1 2765 | . . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V |
5 | 2, 4 | nfim 1813 | . . 3 ⊢ Ⅎ𝑦(∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → 𝐴 ∈ V) |
6 | opprc1 4363 | . . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V → 〈𝐴, 𝑦〉 = ∅) | |
7 | 6 | eleq1d 2672 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (〈𝐴, 𝑦〉 ∈ 𝑉 ↔ ∅ ∈ 𝑉)) |
8 | ax-5 1827 | . . . . . . . . 9 ⊢ (∅ ∈ 𝑉 → ∀𝑦∅ ∈ 𝑉) | |
9 | alneu 39850 | . . . . . . . . 9 ⊢ (∀𝑦∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉) | |
10 | 8, 9 | syl 17 | . . . . . . . 8 ⊢ (∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉) |
11 | 7, 10 | syl6bi 242 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (〈𝐴, 𝑦〉 ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)) |
12 | 11 | impcom 445 | . . . . . 6 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦∅ ∈ 𝑉) |
13 | 7 | eubidv 2478 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 ↔ ∃!𝑦∅ ∈ 𝑉)) |
14 | 13 | notbid 307 | . . . . . . 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 246 | . . . . 5 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉) |
17 | 16 | ex 449 | . . . 4 ⊢ (〈𝐴, 𝑦〉 ∈ 𝑉 → (¬ 𝐴 ∈ V → ¬ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉)) |
18 | 17 | con4d 113 | . . 3 ⊢ (〈𝐴, 𝑦〉 ∈ 𝑉 → (∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → 𝐴 ∈ V)) |
19 | 5, 18 | exlimi 2073 | . 2 ⊢ (∃𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → (∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → 𝐴 ∈ V)) |
20 | 1, 19 | mpcom 37 | 1 ⊢ (∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 ∃wex 1695 ∈ wcel 1977 ∃!weu 2458 Vcvv 3173 ∅c0 3874 〈cop 4131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 ax-pow 4769 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-op 4132 |
This theorem is referenced by: afveu 39882 tz6.12-afv 39902 |
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