Theorem sniota 6551

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 2587	. 2 ⊢ Ⅎ𝑥∃!𝑥𝜑
2		nfab1 2906	. 2 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
3		nfiota1 6515	. . 3 ⊢ Ⅎ𝑥(℩𝑥𝜑)
4	3	nfsn 4706	. 2 ⊢ Ⅎ𝑥{(℩𝑥𝜑)}
5		iota1 6537	. . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
6		eqcom 2743	. . . 4 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑))
7	5, 6	bitrdi 287	. . 3 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑)))
8		abid 2717	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
9		velsn 4641	. . 3 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
10	7, 8, 9	3bitr4g 314	. 2 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
11	1, 2, 4, 10	eqrd 4002	1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ∃!weu 2567  {cab 2713  {csn 4625  ℩cio 6511

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-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-v 3481  df-un 3955  df-ss 3967  df-sn 4626  df-pr 4628  df-uni 4907  df-iota 6513

This theorem is referenced by:  snriota  7422