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Theorem eu2ndop1stv 45833
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 2572 . 2 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉 → ∃𝑦𝐴, 𝑦⟩ ∈ 𝑉)
2 nfeu1 2583 . . . 4 𝑦∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉
3 nfcv 2904 . . . . 5 𝑦𝐴
43nfel1 2920 . . . 4 𝑦 𝐴 ∈ V
52, 4nfim 1900 . . 3 𝑦(∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V)
6 opprc1 4898 . . . . . . . . 9 𝐴 ∈ V → ⟨𝐴, 𝑦⟩ = ∅)
76eleq1d 2819 . . . . . . . 8 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ∅ ∈ 𝑉))
8 ax-5 1914 . . . . . . . . 9 (∅ ∈ 𝑉 → ∀𝑦∅ ∈ 𝑉)
9 alneu 45832 . . . . . . . . 9 (∀𝑦∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)
108, 9syl 17 . . . . . . . 8 (∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)
117, 10syl6bi 253 . . . . . . 7 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉))
1211impcom 409 . . . . . 6 ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦∅ ∈ 𝑉)
137eubidv 2581 . . . . . . . 8 𝐴 ∈ V → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉 ↔ ∃!𝑦∅ ∈ 𝑉))
1413notbid 318 . . . . . . 7 𝐴 ∈ V → (¬ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉))
1514adantl 483 . . . . . 6 ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → (¬ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉))
1612, 15mpbird 257 . . . . 5 ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉)
1716ex 414 . . . 4 (⟨𝐴, 𝑦⟩ ∈ 𝑉 → (¬ 𝐴 ∈ V → ¬ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉))
1817con4d 115 . . 3 (⟨𝐴, 𝑦⟩ ∈ 𝑉 → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V))
195, 18exlimi 2211 . 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 205  wa 397  wal 1540  wex 1782  wcel 2107  ∃!weu 2563  Vcvv 3475  c0 4323  cop 4635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-nul 5307  ax-pow 5364
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-op 4636
This theorem is referenced by:  afveu  45861  tz6.12-afv  45881  tz6.12-afv2  45948
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