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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 2655 | . 2 ⊢ (∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → ∃𝑦〈𝐴, 𝑦〉 ∈ 𝑉) | |
2 | nfeu1 2667 | . . . 4 ⊢ Ⅎ𝑦∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 | |
3 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
4 | 3 | nfel1 2991 | . . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V |
5 | 2, 4 | nfim 1888 | . . 3 ⊢ Ⅎ𝑦(∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → 𝐴 ∈ V) |
6 | opprc1 4819 | . . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V → 〈𝐴, 𝑦〉 = ∅) | |
7 | 6 | eleq1d 2894 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (〈𝐴, 𝑦〉 ∈ 𝑉 ↔ ∅ ∈ 𝑉)) |
8 | ax-5 1902 | . . . . . . . . 9 ⊢ (∅ ∈ 𝑉 → ∀𝑦∅ ∈ 𝑉) | |
9 | alneu 43200 | . . . . . . . . 9 ⊢ (∀𝑦∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉) | |
10 | 8, 9 | syl 17 | . . . . . . . 8 ⊢ (∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉) |
11 | 7, 10 | syl6bi 254 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (〈𝐴, 𝑦〉 ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)) |
12 | 11 | impcom 408 | . . . . . 6 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦∅ ∈ 𝑉) |
13 | 7 | eubidv 2665 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 ↔ ∃!𝑦∅ ∈ 𝑉)) |
14 | 13 | notbid 319 | . . . . . . 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 258 | . . . . 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 2207 | . 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 207 ∧ wa 396 ∀wal 1526 ∃wex 1771 ∈ wcel 2105 ∃!weu 2646 Vcvv 3492 ∅c0 4288 〈cop 4563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 ax-pow 5257 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-dif 3936 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-op 4564 |
This theorem is referenced by: afveu 43229 tz6.12-afv 43249 tz6.12-afv2 43316 |
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