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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 2642 | . 2 ⊢ (∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → ∃𝑦〈𝐴, 𝑦〉 ∈ 𝑉) | |
2 | nfeu1 2628 | . . . 4 ⊢ Ⅎ𝑦∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 | |
3 | nfcv 2913 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
4 | 3 | nfel1 2928 | . . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V |
5 | 2, 4 | nfim 1977 | . . 3 ⊢ Ⅎ𝑦(∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → 𝐴 ∈ V) |
6 | opprc1 4564 | . . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V → 〈𝐴, 𝑦〉 = ∅) | |
7 | 6 | eleq1d 2835 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (〈𝐴, 𝑦〉 ∈ 𝑉 ↔ ∅ ∈ 𝑉)) |
8 | ax-5 1991 | . . . . . . . . 9 ⊢ (∅ ∈ 𝑉 → ∀𝑦∅ ∈ 𝑉) | |
9 | alneu 41716 | . . . . . . . . 9 ⊢ (∀𝑦∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉) | |
10 | 8, 9 | syl 17 | . . . . . . . 8 ⊢ (∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉) |
11 | 7, 10 | syl6bi 243 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (〈𝐴, 𝑦〉 ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)) |
12 | 11 | impcom 394 | . . . . . 6 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦∅ ∈ 𝑉) |
13 | 7 | eubidv 2638 | . . . . . . . 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 467 | . . . . . 6 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → (¬ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉)) |
16 | 12, 15 | mpbird 247 | . . . . 5 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉) |
17 | 16 | ex 397 | . . . 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 2242 | . 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 196 ∧ wa 382 ∀wal 1629 ∃wex 1852 ∈ wcel 2145 ∃!weu 2618 Vcvv 3351 ∅c0 4063 〈cop 4323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-nul 4924 ax-pow 4975 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-v 3353 df-dif 3726 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-op 4324 |
This theorem is referenced by: afveu 41748 tz6.12-afv 41768 |
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