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Theorem eu2ndop1stv 47142
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 2576 . 2 (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉 → ∃𝑦𝐴, 𝑦⟩ ∈ 𝑉)
2 nfeu1 2587 . . . 4 𝑦∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉
3 nfcv 2904 . . . . 5 𝑦𝐴
43nfel1 2921 . . . 4 𝑦 𝐴 ∈ V
52, 4nfim 1895 . . 3 𝑦(∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V)
6 opprc1 4896 . . . . . . . . 9 𝐴 ∈ V → ⟨𝐴, 𝑦⟩ = ∅)
76eleq1d 2825 . . . . . . . 8 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ∅ ∈ 𝑉))
8 ax-5 1909 . . . . . . . . 9 (∅ ∈ 𝑉 → ∀𝑦∅ ∈ 𝑉)
9 alneu 47141 . . . . . . . . 9 (∀𝑦∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)
108, 9syl 17 . . . . . . . 8 (∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)
117, 10biimtrdi 253 . . . . . . 7 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉))
1211impcom 407 . . . . . 6 ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦∅ ∈ 𝑉)
137eubidv 2585 . . . . . . . 8 𝐴 ∈ V → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉 ↔ ∃!𝑦∅ ∈ 𝑉))
1413notbid 318 . . . . . . 7 𝐴 ∈ V → (¬ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉))
1514adantl 481 . . . . . 6 ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → (¬ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉))
1612, 15mpbird 257 . . . . 5 ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉)
1716ex 412 . . . 4 (⟨𝐴, 𝑦⟩ ∈ 𝑉 → (¬ 𝐴 ∈ V → ¬ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉))
1817con4d 115 . . 3 (⟨𝐴, 𝑦⟩ ∈ 𝑉 → (∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V))
195, 18exlimi 2216 . 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 206  wa 395  wal 1537  wex 1778  wcel 2107  ∃!weu 2567  Vcvv 3479  c0 4332  cop 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-nul 5305  ax-pow 5364
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-dif 3953  df-ss 3967  df-nul 4333  df-if 4525  df-op 4632
This theorem is referenced by:  afveu  47170  tz6.12-afv  47190  tz6.12-afv2  47257
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