Theorem euuni 1954

Description: If φ is true for exactly one x, then ∪{x∣φ} is a way to express 'the unique element such that'. Some books use a special symbol such as iota to denote 'the unique element such that'.

Assertion

Ref	Expression
euuni	⊢ (∃!xφ → (φ ↔ ∪{x∣φ} = x))

Proof of Theorem euuni

Step	Hyp	Ref	Expression
1		euabex 1869	. . . 4 ⊢ (∃!xφ → {x∣φ} ∈ V)
2		uniexg 1948	. . . 4 ⊢ ({x∣φ} ∈ V → ∪{x∣φ} ∈ V)
3	1, 2	syl 12	. . 3 ⊢ (∃!xφ → ∪{x∣φ} ∈ V)
4		eueq 1427	. . . 4 ⊢ (∪{x∣φ} ∈ V ↔ ∃!y y = ∪{x∣φ})
5		cleqcom 1103	. . . . 5 ⊢ (y = ∪{x∣φ} ↔ ∪{x∣φ} = y)
6	5	bieu 1014	. . . 4 ⊢ (∃!y y = ∪{x∣φ} ↔ ∃!y∪{x∣φ} = y)
7		hbab1 1095	. . . . . . 7 ⊢ (z ∈ {x∣φ} → ∀x z ∈ {x∣φ})
8	7	hbuni 1925	. . . . . 6 ⊢ (z ∈ ∪{x∣φ} → ∀x z ∈ ∪{x∣φ})
9		ax-17 925	. . . . . 6 ⊢ (z ∈ y → ∀x z ∈ y)
10	8, 9	hbeq 1171	. . . . 5 ⊢ (∪{x∣φ} = y → ∀x∪{x∣φ} = y)
11		ax-17 925	. . . . 5 ⊢ (∪{x∣φ} = x → ∀y∪{x∣φ} = x)
12		cleq2 1110	. . . . 5 ⊢ (y = x → (∪{x∣φ} = y ↔ ∪{x∣φ} = x))
13	10, 11, 12	cbveu 1018	. . . 4 ⊢ (∃!y∪{x∣φ} = y ↔ ∃!x∪{x∣φ} = x)
14	4, 6, 13	3bitr 155	. . 3 ⊢ (∪{x∣φ} ∈ V ↔ ∃!x∪{x∣φ} = x)
15	3, 14	sylib 173	. 2 ⊢ (∃!xφ → ∃!x∪{x∣φ} = x)
16		eusn 1913	. . 3 ⊢ (∃!xφ ↔ ∃x{x∣φ} = {x})
17		visset 1350	. . . . . . . 8 ⊢ x ∈ V
18	17	snid 1830	. . . . . . 7 ⊢ x ∈ {x}
19		eleq2 1150	. . . . . . 7 ⊢ ({x∣φ} = {x} → (x ∈ {x∣φ} ↔ x ∈ {x}))
20	18, 19	mpbiri 169	. . . . . 6 ⊢ ({x∣φ} = {x} → x ∈ {x∣φ})
21		abid 1094	. . . . . 6 ⊢ (x ∈ {x∣φ} ↔ φ)
22	20, 21	sylib 173	. . . . 5 ⊢ ({x∣φ} = {x} → φ)
23		unieq 1927	. . . . . 6 ⊢ ({x∣φ} = {x} → ∪{x∣φ} = ∪{x})
24	17	unisn 1932	. . . . . 6 ⊢ ∪{x} = x
25	23, 24	syl6eq 1140	. . . . 5 ⊢ ({x∣φ} = {x} → ∪{x∣φ} = x)
26	22, 25	jca 236	. . . 4 ⊢ ({x∣φ} = {x} → (φ ∧ ∪{x∣φ} = x))
27	26	19.22i 723	. . 3 ⊢ (∃x{x∣φ} = {x} → ∃x(φ ∧ ∪{x∣φ} = x))
28	16, 27	sylbi 174	. 2 ⊢ (∃!xφ → ∃x(φ ∧ ∪{x∣φ} = x))
29		eupickb 1056	. . . 4 ⊢ ((∃!xφ ∧ ∃!x∪{x∣φ} = x ∧ ∃x(φ ∧ ∪{x∣φ} = x)) → (φ ↔ ∪{x∣φ} = x))
30	29	3exp 611	. . 3 ⊢ (∃!xφ → (∃!x∪{x∣φ} = x → (∃x(φ ∧ ∪{x∣φ} = x) → (φ ↔ ∪{x∣φ} = x))))
31	30	imp3a 279	. 2 ⊢ (∃!xφ → ((∃!x∪{x∣φ} = x ∧ ∃x(φ ∧ ∪{x∣φ} = x)) → (φ ↔ ∪{x∣φ} = x)))
32	15, 28, 31	mp2and 526	1 ⊢ (∃!xφ → (φ ↔ ∪{x∣φ} = x))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   ∈ wel 803  ∃!weu 1007  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808  ∪cuni 1919

This theorem is referenced by:  reuuni1 1955

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-uni 1920

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