Theorem sniota 6564

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 2591	. 2 ⊢ Ⅎ𝑥∃!𝑥𝜑
2		nfab1 2910	. 2 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
3		nfiota1 6527	. . 3 ⊢ Ⅎ𝑥(℩𝑥𝜑)
4	3	nfsn 4732	. 2 ⊢ Ⅎ𝑥{(℩𝑥𝜑)}
5		iota1 6550	. . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
6		eqcom 2747	. . . 4 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑))
7	5, 6	bitrdi 287	. . 3 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑)))
8		abid 2721	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
9		velsn 4664	. . 3 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
10	7, 8, 9	3bitr4g 314	. 2 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
11	1, 2, 4, 10	eqrd 4028	1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ∃!weu 2571  {cab 2717  {csn 4648  ℩cio 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-v 3490  df-un 3981  df-ss 3993  df-sn 4649  df-pr 4651  df-uni 4932  df-iota 6525

This theorem is referenced by:  snriota  7438