Theorem reuun1 4251

Description: Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)

Assertion

Ref	Expression
reuun1	⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴 ∪ 𝐵)(𝜑 ∨ 𝜓)) → ∃!𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem reuun1

Step	Hyp	Ref	Expression
1		ssun1 4106	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		orc 864	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
3	2	rgenw 3076	. 2 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → (𝜑 ∨ 𝜓))
4		reuss2 4249	. 2 ⊢ (((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝜑 → (𝜑 ∨ 𝜓))) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴 ∪ 𝐵)(𝜑 ∨ 𝜓))) → ∃!𝑥 ∈ 𝐴 𝜑)
5	1, 3, 4	mpanl12 699	1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴 ∪ 𝐵)(𝜑 ∨ 𝜓)) → ∃!𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844  ∀wral 3064  ∃wrex 3065  ∃!wreu 3066   ∪ cun 3885   ⊆ wss 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-reu 3072  df-v 3434  df-un 3892  df-in 3894  df-ss 3904

This theorem is referenced by: (None)