Theorem eu2ndop1stv 45918

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	⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V)

Distinct variable groups:   𝑦,𝐴   𝑦,𝑉

Proof of Theorem eu2ndop1stv

Step	Hyp	Ref	Expression
1		euex 2571	. 2 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉)
2		nfeu1 2582	. . . 4 ⊢ Ⅎ𝑦∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉
3		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑦𝐴
4	3	nfel1 2919	. . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V
5	2, 4	nfim 1899	. . 3 ⊢ Ⅎ𝑦(∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V)
6		opprc1 4897	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝑦⟩ = ∅)
7	6	eleq1d 2818	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ∅ ∈ 𝑉))
8		ax-5 1913	. . . . . . . . 9 ⊢ (∅ ∈ 𝑉 → ∀𝑦∅ ∈ 𝑉)
9		alneu 45917	. . . . . . . . 9 ⊢ (∀𝑦∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)
10	8, 9	syl 17	. . . . . . . 8 ⊢ (∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)
11	7, 10	syl6bi 252	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉))
12	11	impcom 408	. . . . . 6 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦∅ ∈ 𝑉)
13	7	eubidv 2580	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ∃!𝑦∅ ∈ 𝑉))
14	13	notbid 317	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (¬ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉))
15	14	adantl 482	. . . . . 6 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → (¬ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉))
16	12, 15	mpbird 256	. . . . 5 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉)
17	16	ex 413	. . . 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 2210	. 2 ⊢ (∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V))
20	1, 19	mpcom 38	1 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539  ∃wex 1781   ∈ wcel 2106  ∃!weu 2562  Vcvv 3474  ∅c0 4322  ⟨cop 4634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-nul 5306  ax-pow 5363

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-v 3476  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-op 4635

This theorem is referenced by:  afveu  45946  tz6.12-afv  45966  tz6.12-afv2  46033