Theorem reuuni3 1958

Description: Derive the property χ of 'the unique element in A such that φ ' when expressed explicitly as ∪{y ∈ A∣ψ}.

Hypotheses

Ref	Expression
reuuni3.1	⊢ (x = y → (φ ↔ ψ))
reuuni3.2	⊢ (x = ∪{y ∈ A∣ψ} → (φ ↔ χ))

Assertion

Ref	Expression
reuuni3	⊢ (∃!x ∈ A φ → χ)

Distinct variable group(s):   x,y,A   φ,y   ψ,x   χ,x

Proof of Theorem reuuni3

Step	Hyp	Ref	Expression
1		reucl 1957	. . 3 ⊢ (∃!x ∈ A φ → ∪{x ∈ A∣φ} ∈ A)
2		reuuni3.1	. . . . . 6 ⊢ (x = y → (φ ↔ ψ))
3	2	cbvrabv 1426	. . . . 5 ⊢ {x ∈ A∣φ} = {y ∈ A∣ψ}
4	3	unieqi 1928	. . . 4 ⊢ ∪{x ∈ A∣φ} = ∪{y ∈ A∣ψ}
5	4	eleq1i 1152	. . 3 ⊢ (∪{x ∈ A∣φ} ∈ A ↔ ∪{y ∈ A∣ψ} ∈ A)
6	1, 5	sylib 173	. 2 ⊢ (∃!x ∈ A φ → ∪{y ∈ A∣ψ} ∈ A)
7		reuuni3.2	. . . 4 ⊢ (x = ∪{y ∈ A∣ψ} → (φ ↔ χ))
8	7	reuuni2 1956	. . 3 ⊢ ((∪{y ∈ A∣ψ} ∈ A ∧ ∃!x ∈ A φ) → (χ ↔ ∪{x ∈ A∣φ} = ∪{y ∈ A∣ψ}))
9	4, 8	mpbiri 169	. 2 ⊢ ((∪{y ∈ A∣ψ} ∈ A ∧ ∃!x ∈ A φ) → χ)
10	6, 9	mpancom 528	1 ⊢ (∃!x ∈ A φ → χ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∃!wreu 1203  {crab 1204  ∪cuni 1919

This theorem is referenced by:  uzwo3lem2 4615  flleltt 4625

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-reu 1207  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-uni 1920

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