Theorem reuun1 3538

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 3427	. 2 ⊢ A ⊆ (A ∪ B)
2		orc 374	. . 3 ⊢ (φ → (φ ∨ ψ))
3	2	rgenw 2682	. 2 ⊢ ∀x ∈ A (φ → (φ ∨ ψ))
4		reuss2 3536	. 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 663	1 ⊢ ((∃x ∈ A φ ∧ ∃!x ∈ (A ∪ B)(φ ∨ ψ)) → ∃!x ∈ A φ)

Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358  ∀wral 2615  ∃wrex 2616  ∃!wreu 2617   ∪ cun 3208   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-reu 2622  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-ss 3260

This theorem is referenced by: (None)