Theorem eu2ndop1stv 47807

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 2603	. 2 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉)
2		nfeu1 2615	. . . 4 ⊢ Ⅎ𝑦∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉
3		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑦𝐴
4	3	nfel1 2939	. . . 4 ⊢ Ⅎ𝑦 𝐴 ∈ V
5	2, 4	nfim 1924	. . 3 ⊢ Ⅎ𝑦(∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V)
6		opprc1 4861	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝑦⟩ = ∅)
7	6	eleq1d 2846	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (⟨𝐴, 𝑦⟩ ∈ 𝑉 ↔ ∅ ∈ 𝑉))
8		ax-5 1938	. . . . . . . . 9 ⊢ (∅ ∈ 𝑉 → ∀𝑦∅ ∈ 𝑉)
9		alneu 47806	. . . . . . . . 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 412	. . . . . 6 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝑉 ∧ ¬ 𝐴 ∈ V) → ¬ ∃!𝑦∅ ∈ 𝑉)
13	7	eubidv 2612	. . . . . . . 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 486	. . . . . 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 417	. . . 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 2251	. 2 ⊢ (∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V))
20	1, 19	mpcom 39	1 ⊢ (∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1566  ∃wex 1807   ∈ wcel 2141  ∃!weu 2594  Vcvv 3453  ∅c0 4285  ⟨cop 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-nul 5268  ax-pow 5336

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-dif 3907  df-ss 3921  df-nul 4286  df-if 4487  df-op 4595

This theorem is referenced by:  afveu  47835  tz6.12-afv  47855  tz6.12-afv2  47922