Theorem sniota 6468

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 2582	. 2 ⊢ Ⅎ𝑥∃!𝑥𝜑
2		nfab1 2894	. 2 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
3		nfiota1 6435	. . 3 ⊢ Ⅎ𝑥(℩𝑥𝜑)
4	3	nfsn 4658	. 2 ⊢ Ⅎ𝑥{(℩𝑥𝜑)}
5		iota1 6456	. . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
6		eqcom 2737	. . . 4 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑))
7	5, 6	bitrdi 287	. . 3 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑)))
8		abid 2712	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
9		velsn 4590	. . 3 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
10	7, 8, 9	3bitr4g 314	. 2 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
11	1, 2, 4, 10	eqrd 3952	1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)})

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2110  ∃!weu 2562  {cab 2708  {csn 4574  ℩cio 6431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-v 3436  df-un 3905  df-ss 3917  df-sn 4575  df-pr 4577  df-uni 4858  df-iota 6433

This theorem is referenced by:  snriota  7331