Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 2658 | . 2 ⊢ (∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → ∃𝑦〈𝐴, 𝑦〉 ∈ 𝑉) | |
2 | nfeu1 2670 | . . . 4 ⊢ Ⅎ𝑦∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 | |
3 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
4 | 3 | nfel1 2994 | . . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V |
5 | 2, 4 | nfim 1893 | . . 3 ⊢ Ⅎ𝑦(∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 → 𝐴 ∈ V) |
6 | opprc1 4826 | . . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V → 〈𝐴, 𝑦〉 = ∅) | |
7 | 6 | eleq1d 2897 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (〈𝐴, 𝑦〉 ∈ 𝑉 ↔ ∅ ∈ 𝑉)) |
8 | ax-5 1907 | . . . . . . . . 9 ⊢ (∅ ∈ 𝑉 → ∀𝑦∅ ∈ 𝑉) | |
9 | alneu 43322 | . . . . . . . . 9 ⊢ (∀𝑦∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉) | |
10 | 8, 9 | syl 17 | . . . . . . . 8 ⊢ (∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉) |
11 | 7, 10 | syl6bi 255 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (〈𝐴, 𝑦〉 ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)) |
12 | 11 | impcom 410 | . . . . . 6 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦∅ ∈ 𝑉) |
13 | 7 | eubidv 2668 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 ↔ ∃!𝑦∅ ∈ 𝑉)) |
14 | 13 | notbid 320 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (¬ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉)) |
15 | 14 | adantl 484 | . . . . . 6 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → (¬ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉)) |
16 | 12, 15 | mpbird 259 | . . . . 5 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝑉) |
17 | 16 | ex 415 | . . . 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 2213 | . 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 208 ∧ wa 398 ∀wal 1531 ∃wex 1776 ∈ wcel 2110 ∃!weu 2649 Vcvv 3494 ∅c0 4290 〈cop 4572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5209 ax-pow 5265 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-dif 3938 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-op 4573 |
This theorem is referenced by: afveu 43351 tz6.12-afv 43371 tz6.12-afv2 43438 |
Copyright terms: Public domain | W3C validator |