Theorem reuun1 4327

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 4177	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		orc 867	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
3	2	rgenw 3064	. 2 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → (𝜑 ∨ 𝜓))
4		reuss2 4325	. 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 702	1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴 ∪ 𝐵)(𝜑 ∨ 𝜓)) → ∃!𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847  ∀wral 3060  ∃wrex 3069  ∃!wreu 3377   ∪ cun 3948   ⊆ wss 3950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-reu 3380  df-v 3481  df-un 3955  df-ss 3967

This theorem is referenced by: (None)