Theorem sniota 6409

Description: A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
sniota	⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)})

Proof of Theorem sniota

Step	Hyp	Ref	Expression
1		nfeu1 2588	. 2 ⊢ Ⅎ𝑥∃!𝑥𝜑
2		nfab1 2908	. 2 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
3		nfiota1 6378	. . 3 ⊢ Ⅎ𝑥(℩𝑥𝜑)
4	3	nfsn 4640	. 2 ⊢ Ⅎ𝑥{(℩𝑥𝜑)}
5		iota1 6395	. . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
6		eqcom 2745	. . . 4 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑))
7	5, 6	bitrdi 286	. . 3 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑)))
8		abid 2719	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
9		velsn 4574	. . 3 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
10	7, 8, 9	3bitr4g 313	. 2 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
11	1, 2, 4, 10	eqrd 3936	1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ∃!weu 2568  {cab 2715  {csn 4558  ℩cio 6374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-v 3424  df-un 3888  df-in 3890  df-ss 3900  df-sn 4559  df-pr 4561  df-uni 4837  df-iota 6376

This theorem is referenced by:  snriota  7246