Theorem sniota 4370

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 2214	. . 3 ⊢ Ⅎx∃!xφ
2		iota1 4354	. . . . 5 ⊢ (∃!xφ → (φ ↔ (℩xφ) = x))
3		eqcom 2355	. . . . 5 ⊢ ((℩xφ) = x ↔ x = (℩xφ))
4	2, 3	syl6bb 252	. . . 4 ⊢ (∃!xφ → (φ ↔ x = (℩xφ)))
5		abid 2341	. . . 4 ⊢ (x ∈ {x ∣ φ} ↔ φ)
6		vex 2863	. . . . 5 ⊢ x ∈ V
7	6	elsnc 3757	. . . 4 ⊢ (x ∈ {(℩xφ)} ↔ x = (℩xφ))
8	4, 5, 7	3bitr4g 279	. . 3 ⊢ (∃!xφ → (x ∈ {x ∣ φ} ↔ x ∈ {(℩xφ)}))
9	1, 8	alrimi 1765	. 2 ⊢ (∃!xφ → ∀x(x ∈ {x ∣ φ} ↔ x ∈ {(℩xφ)}))
10		nfab1 2492	. . 3 ⊢ Ⅎx{x ∣ φ}
11		nfiota1 4342	. . . 4 ⊢ Ⅎx(℩xφ)
12	11	nfsn 3785	. . 3 ⊢ Ⅎx{(℩xφ)}
13	10, 12	cleqf 2514	. 2 ⊢ ({x ∣ φ} = {(℩xφ)} ↔ ∀x(x ∈ {x ∣ φ} ↔ x ∈ {(℩xφ)}))
14	9, 13	sylibr 203	1 ⊢ (∃!xφ → {x ∣ φ} = {(℩xφ)})

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  {csn 3738  ℩cio 4338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-uni 3893  df-iota 4340

This theorem is referenced by: (None)