Theorem eu2ndop1stv 48195

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 2602	. 2 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉)
2		nfeu1 2614	. . . 4 ⊢ Ⅎ𝑦∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉
3		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑦𝐴
4	3	nfel1 2938	. . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V
5	2, 4	nfim 1928	. . 3 ⊢ Ⅎ𝑦(∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V)
6		opprc1 4856	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝑦⟩ = ∅)
7	6	eleq1d 2845	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ∅ ∈ 𝑉))
8		ax-5 1942	. . . . . . . . 9 ⊢ (∅ ∈ 𝑉 → ∀𝑦∅ ∈ 𝑉)
9		alneu 48194	. . . . . . . . 9 ⊢ (∀𝑦∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)
10	8, 9	syl 18	. . . . . . . 8 ⊢ (∅ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉)
11	7, 10	biimtrdi 256	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 → ¬ ∃!𝑦∅ ∈ 𝑉))
12	11	impcom 413	. . . . . 6 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦∅ ∈ 𝑉)
13	7	eubidv 2611	. . . . . . . 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 487	. . . . . 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 418	. . . 4 ⊢ (⟨𝐴, 𝑦⟩ ∈ 𝑉 → (¬ 𝐴 ∈ V → ¬ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉))
18	17	con4d 116	. . 3 ⊢ (⟨𝐴, 𝑦⟩ ∈ 𝑉 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V))
19	5, 18	exlimi 2253	. 2 ⊢ (∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V))
20	1, 19	mpcom 39	1 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567  ∃wex 1811   ∈ wcel 2144  ∃!weu 2593  Vcvv 3450  ∅c0 4278  ⟨cop 4589

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-nul 5259  ax-pow 5326

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ss 3896  df-dif 4047  df-nul 4279  df-if 4482  df-op 4590

This theorem is used by:  afveu  48223  tz6.12-afv  48243  tz6.12-afv2  48310