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Theorem eu2ndop1stv 39645
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 2482 . 2 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉 → ∃𝑦𝐴, 𝑦⟩ ∈ 𝑉)
2 nfeu1 2468 . . . 4 𝑦∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉
3 nfcv 2751 . . . . 5 𝑦𝐴
43nfel1 2765 . . . 4 𝑦 𝐴 ∈ V
52, 4nfim 1813 . . 3 𝑦(∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V)
6 opprc1 4358 . . . . . . . . 9 𝐴 ∈ V → ⟨𝐴, 𝑦⟩ = ∅)
76eleq1d 2672 . . . . . . . 8 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ∅ ∈ 𝑉))
8 ax-5 1827 . . . . . . . . 9 (∅ ∈ 𝑉 → ∀𝑦∅ ∈ 𝑉)
9 alneu 39644 . . . . . . . . 9 (∀𝑦∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)
108, 9syl 17 . . . . . . . 8 (∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)
117, 10syl6bi 242 . . . . . . 7 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉))
1211impcom 445 . . . . . 6 ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦∅ ∈ 𝑉)
137eubidv 2478 . . . . . . . 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 246 . . . . 5 ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉)
1716ex 449 . . . 4 (⟨𝐴, 𝑦⟩ ∈ 𝑉 → (¬ 𝐴 ∈ V → ¬ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉))
1817con4d 113 . . 3 (⟨𝐴, 𝑦⟩ ∈ 𝑉 → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V))
195, 18exlimi 2073 . 2 (∃𝑦𝐴, 𝑦⟩ ∈ 𝑉 → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V))
201, 19mpcom 37 1 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ 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 4712  ax-pow 4764
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  39677  tz6.12-afv  39697
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