Theorem reiotacl 4365

Description: Membership law for descriptions. (Contributed by SF, 21-Aug-2011.)

Assertion

Ref	Expression
reiotacl	⊢ (∃!x ∈ A φ → (℩x(x ∈ A ∧ φ)) ∈ A)

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem reiotacl

Step	Hyp	Ref	Expression
1		ssrab2 3352	. . 3 ⊢ {x ∈ A ∣ φ} ⊆ A
2	1	a1i 10	. 2 ⊢ (∃!x ∈ A φ → {x ∈ A ∣ φ} ⊆ A)
3		reiotacl2 4364	. 2 ⊢ (∃!x ∈ A φ → (℩x(x ∈ A ∧ φ)) ∈ {x ∈ A ∣ φ})
4	2, 3	sseldd 3275	1 ⊢ (∃!x ∈ A φ → (℩x(x ∈ A ∧ φ)) ∈ A)

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∃!wreu 2617  {crab 2619   ⊆ wss 3258  ℩cio 4338

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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-reu 2622  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-ss 3260  df-sn 3742  df-pr 3743  df-uni 3893  df-iota 4340

This theorem is referenced by:  ncfinprop  4475  tfinprop  4490  tccl  6161