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Theorem eu2ndop1stv 41523
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.)
Assertion
Ref Expression
eu2ndop1stv (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V)
Distinct variable groups:   𝑦,𝐴   𝑦,𝑉

Proof of Theorem eu2ndop1stv
StepHypRef Expression
1 euex 2522 . 2 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉 → ∃𝑦𝐴, 𝑦⟩ ∈ 𝑉)
2 nfeu1 2508 . . . 4 𝑦∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉
3 nfcv 2793 . . . . 5 𝑦𝐴
43nfel1 2808 . . . 4 𝑦 𝐴 ∈ V
52, 4nfim 1865 . . 3 𝑦(∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V)
6 opprc1 4457 . . . . . . . . 9 𝐴 ∈ V → ⟨𝐴, 𝑦⟩ = ∅)
76eleq1d 2715 . . . . . . . 8 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ∅ ∈ 𝑉))
8 ax-5 1879 . . . . . . . . 9 (∅ ∈ 𝑉 → ∀𝑦∅ ∈ 𝑉)
9 alneu 41522 . . . . . . . . 9 (∀𝑦∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)
108, 9syl 17 . . . . . . . 8 (∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)
117, 10syl6bi 243 . . . . . . 7 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉))
1211impcom 445 . . . . . 6 ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦∅ ∈ 𝑉)
137eubidv 2518 . . . . . . . 8 𝐴 ∈ V → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉 ↔ ∃!𝑦∅ ∈ 𝑉))
1413notbid 307 . . . . . . 7 𝐴 ∈ V → (¬ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉))
1514adantl 481 . . . . . 6 ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → (¬ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉))
1612, 15mpbird 247 . . . . 5 ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉)
1716ex 449 . . . 4 (⟨𝐴, 𝑦⟩ ∈ 𝑉 → (¬ 𝐴 ∈ V → ¬ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉))
1817con4d 114 . . 3 (⟨𝐴, 𝑦⟩ ∈ 𝑉 → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V))
195, 18exlimi 2124 . 2 (∃𝑦𝐴, 𝑦⟩ ∈ 𝑉 → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V))
201, 19mpcom 38 1 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521  wex 1744  wcel 2030  ∃!weu 2498  Vcvv 3231  c0 3948  cop 4216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822  ax-pow 4873
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-op 4217
This theorem is referenced by:  afveu  41554  tz6.12-afv  41574
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