Theorem reuuni3 2892

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 groups:   x,y,A   φ,y   ψ,x   χ,x

Proof of Theorem reuuni3

Step	Hyp	Ref	Expression
1		reucl 2891	. . 3 ⊢ (∃!x ∈ A φ → ∪{x ∈ A∣φ} ∈ A)
2		reuuni3.1	. . . . 5 ⊢ (x = y → (φ ↔ ψ))
3	2	cbvrabv 1914	. . . 4 ⊢ {x ∈ A∣φ} = {y ∈ A∣ψ}
4	3	unieqi 2515	. . 3 ⊢ ∪{x ∈ A∣φ} = ∪{y ∈ A∣ψ}
5	1, 4	syl5eqelr 1556	. 2 ⊢ (∃!x ∈ A φ → ∪{y ∈ A∣ψ} ∈ A)
6		reuuni3.2	. . . 4 ⊢ (x = ∪{y ∈ A∣ψ} → (φ ↔ χ))
7	6	reuuni2 2890	. . 3 ⊢ ((∪{y ∈ A∣ψ} ∈ A ⋀ ∃!x ∈ A φ) → (χ ↔ ∪{x ∈ A∣φ} = ∪{y ∈ A∣ψ}))
8	4, 7	mpbiri 194	. 2 ⊢ ((∪{y ∈ A∣ψ} ∈ A ⋀ ∃!x ∈ A φ) → χ)
9	5, 8	mpancom 707	1 ⊢ (∃!x ∈ A φ → χ)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∃!wreu 1650  {crab 1651  ∪cuni 2507

This theorem is referenced by:  lble 6049  uzwo3lem2 6219  flleltt 6230

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-reu 1654  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-uni 2508

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