Theorem eu2ndop1stv 44289

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 2576	. 2 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉)
2		nfeu1 2587	. . . 4 ⊢ Ⅎ𝑦∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉
3		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑦𝐴
4	3	nfel1 2920	. . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V
5	2, 4	nfim 1904	. . 3 ⊢ Ⅎ𝑦(∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V)
6		opprc1 4808	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝑦⟩ = ∅)
7	6	eleq1d 2822	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ∅ ∈ 𝑉))
8		ax-5 1918	. . . . . . . . 9 ⊢ (∅ ∈ 𝑉 → ∀𝑦∅ ∈ 𝑉)
9		alneu 44288	. . . . . . . . 9 ⊢ (∀𝑦∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)
10	8, 9	syl 17	. . . . . . . 8 ⊢ (∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)
11	7, 10	syl6bi 256	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉))
12	11	impcom 411	. . . . . 6 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦∅ ∈ 𝑉)
13	7	eubidv 2585	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ∃!𝑦∅ ∈ 𝑉))
14	13	notbid 321	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (¬ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉))
15	14	adantl 485	. . . . . 6 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → (¬ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ¬ ∃!𝑦∅ ∈ 𝑉))
16	12, 15	mpbird 260	. . . . 5 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉)
17	16	ex 416	. . . 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 2215	. 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 209   ∧ wa 399  ∀wal 1541  ∃wex 1787   ∈ wcel 2110  ∃!weu 2567  Vcvv 3408  ∅c0 4237  ⟨cop 4547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-nul 5199  ax-pow 5258

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-v 3410  df-dif 3869  df-nul 4238  df-if 4440  df-op 4548

This theorem is referenced by:  afveu  44317  tz6.12-afv  44337  tz6.12-afv2  44404