Theorem sniota 5209

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 2037	. . 3 ⊢ Ⅎ𝑥∃!𝑥𝜑
2		iota1 5194	. . . . 5 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
3		eqcom 2179	. . . . 5 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑))
4	2, 3	bitrdi 196	. . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑)))
5		abid 2165	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
6		vex 2742	. . . . 5 ⊢ 𝑥 ∈ V
7	6	elsn 3610	. . . 4 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑))
8	4, 5, 7	3bitr4g 223	. . 3 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
9	1, 8	alrimi 1522	. 2 ⊢ (∃!𝑥𝜑 → ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)}))
10		nfab1 2321	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
11		nfiota1 5182	. . . 4 ⊢ Ⅎ𝑥(℩𝑥𝜑)
12	11	nfsn 3654	. . 3 ⊢ Ⅎ𝑥{(℩𝑥𝜑)}
13	10, 12	cleqf 2344	. 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 1351   = wceq 1353  ∃!weu 2026   ∈ wcel 2148  {cab 2163  {csn 3594  ℩cio 5178

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-sn 3600  df-pr 3601  df-uni 3812  df-iota 5180

This theorem is referenced by:  snriota  5862