Theorem sniota 5363

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 2097	. . 3 ⊢ Ⅎ𝑥∃!𝑥𝜑
2		iota1 5347	. . . . 5 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
3		eqcom 2240	. . . . 5 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑))
4	2, 3	bitrdi 196	. . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑)))
5		abid 2226	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
6		vex 2824	. . . . 5 ⊢ 𝑥 ∈ V
7	6	elsn 3721	. . . 4 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
8	4, 5, 7	3bitr4g 223	. . 3 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
9	1, 8	alrimi 1575	. 2 ⊢ (∃!𝑥𝜑 → ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
10		nfab1 2394	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
11		nfiota1 5334	. . . 4 ⊢ Ⅎ𝑥(℩𝑥𝜑)
12	11	nfsn 3765	. . 3 ⊢ Ⅎ𝑥{(℩𝑥𝜑)}
13	10, 12	cleqf 2417	. 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 1400   = wceq 1402  ∃!weu 2086   ∈ wcel 2209  {cab 2224  {csn 3705  ℩cio 5330

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-sn 3711  df-pr 3712  df-uni 3931  df-iota 5332

This theorem is referenced by:  snriota  6060