Theorem sniota 6527

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 2615	. 2 ⊢ Ⅎ𝑥∃!𝑥𝜑
2		nfab1 2925	. 2 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
3		nfiota1 6494	. . 3 ⊢ Ⅎ𝑥(℩𝑥𝜑)
4	3	nfsn 4672	. 2 ⊢ Ⅎ𝑥{(℩𝑥𝜑)}
5		iota1 6515	. . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
6		eqcom 2768	. . . 4 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑))
7	5, 6	bitrdi 290	. . 3 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑)))
8		abid 2743	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
9		velsn 4604	. . 3 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
10	7, 8, 9	3bitr4g 317	. 2 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
11	1, 2, 4, 10	eqrd 3955	1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)})

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2141  ∃!weu 2594  {cab 2739  {csn 4588  ℩cio 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-v 3455  df-un 3909  df-ss 3921  df-sn 4589  df-pr 4591  df-uni 4872  df-iota 6492

This theorem is referenced by:  snriota  7400