Theorem sniota 5343

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 2091	. . 3 ⊢ Ⅎ𝑥∃!𝑥𝜑
2		iota1 5327	. . . . 5 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
3		eqcom 2234	. . . . 5 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑))
4	2, 3	bitrdi 196	. . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑)))
5		abid 2220	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
6		vex 2816	. . . . 5 ⊢ 𝑥 ∈ V
7	6	elsn 3705	. . . 4 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
8	4, 5, 7	3bitr4g 223	. . 3 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
9	1, 8	alrimi 1571	. 2 ⊢ (∃!𝑥𝜑 → ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
10		nfab1 2386	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
11		nfiota1 5314	. . . 4 ⊢ Ⅎ𝑥(℩𝑥𝜑)
12	11	nfsn 3749	. . 3 ⊢ Ⅎ𝑥{(℩𝑥𝜑)}
13	10, 12	cleqf 2409	. 2 ⊢ ({𝑥 ∣ 𝜑} = {(℩𝑥𝜑)} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
14	9, 13	sylibr 134	1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)})

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1396   = wceq 1398  ∃!weu 2080   ∈ wcel 2203  {cab 2218  {csn 3689  ℩cio 5310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-sn 3695  df-pr 3696  df-uni 3915  df-iota 5312

This theorem is referenced by:  snriota  6035