Theorem euuni 2877

Description: If φ is true for exactly one x, then ∪{x∣φ} is a way to express "the unique element such that φ is true." 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 2763	. . . 4 ⊢ (∃!xφ → {x∣φ} ∈ V)
2		uniexg 2867	. . . 4 ⊢ ({x∣φ} ∈ V → ∪{x∣φ} ∈ V)
3	1, 2	syl 10	. . 3 ⊢ (∃!xφ → ∪{x∣φ} ∈ V)
4		eueq 1913	. . . 4 ⊢ (∪{x∣φ} ∈ V ↔ ∃!y y = ∪{x∣φ})
5		eqcom 1475	. . . . 5 ⊢ (y = ∪{x∣φ} ↔ ∪{x∣φ} = y)
6	5	eubii 1386	. . . 4 ⊢ (∃!y y = ∪{x∣φ} ↔ ∃!y∪{x∣φ} = y)
7		hbab1 1465	. . . . . . 7 ⊢ (z ∈ {x∣φ} → ∀x z ∈ {x∣φ})
8	7	hbuni 2505	. . . . . 6 ⊢ (z ∈ ∪{x∣φ} → ∀x z ∈ ∪{x∣φ})
9		ax-17 970	. . . . . 6 ⊢ (z ∈ y → ∀x z ∈ y)
10	8, 9	hbeq 1563	. . . . 5 ⊢ (∪{x∣φ} = y → ∀x∪{x∣φ} = y)
11		ax-17 970	. . . . 5 ⊢ (∪{x∣φ} = x → ∀y∪{x∣φ} = x)
12		eqeq2 1482	. . . . 5 ⊢ (y = x → (∪{x∣φ} = y ↔ ∪{x∣φ} = x))
13	10, 11, 12	cbveu 1390	. . . 4 ⊢ (∃!y∪{x∣φ} = y ↔ ∃!x∪{x∣φ} = x)
14	4, 6, 13	3bitr 177	. . 3 ⊢ (∪{x∣φ} ∈ V ↔ ∃!x∪{x∣φ} = x)
15	3, 14	sylib 198	. 2 ⊢ (∃!xφ → ∃!x∪{x∣φ} = x)
16		eusn 2443	. . 3 ⊢ (∃!xφ ↔ ∃x{x∣φ} = {x})
17		visset 1810	. . . . . . . 8 ⊢ x ∈ V
18	17	snid 2432	. . . . . . 7 ⊢ x ∈ {x}
19		eleq2 1533	. . . . . . 7 ⊢ ({x∣φ} = {x} → (x ∈ {x∣φ} ↔ x ∈ {x}))
20	18, 19	mpbiri 194	. . . . . 6 ⊢ ({x∣φ} = {x} → x ∈ {x∣φ})
21		abid 1464	. . . . . 6 ⊢ (x ∈ {x∣φ} ↔ φ)
22	20, 21	sylib 198	. . . . 5 ⊢ ({x∣φ} = {x} → φ)
23		unieq 2506	. . . . . 6 ⊢ ({x∣φ} = {x} → ∪{x∣φ} = ∪{x})
24	17	unisn 2513	. . . . . 6 ⊢ ∪{x} = x
25	23, 24	syl6eq 1521	. . . . 5 ⊢ ({x∣φ} = {x} → ∪{x∣φ} = x)
26	22, 25	jca 288	. . . 4 ⊢ ({x∣φ} = {x} → (φ ⋀ ∪{x∣φ} = x))
27	26	19.22i 1039	. . 3 ⊢ (∃x{x∣φ} = {x} → ∃x(φ ⋀ ∪{x∣φ} = x))
28	16, 27	sylbi 199	. 2 ⊢ (∃!xφ → ∃x(φ ⋀ ∪{x∣φ} = x))
29		eupickb 1434	. 2 ⊢ ((∃!xφ ⋀ ∃!x∪{x∣φ} = x ⋀ ∃x(φ ⋀ ∪{x∣φ} = x)) → (φ ↔ ∪{x∣φ} = x))
30	15, 28, 29	mpd3an23 917	1 ⊢ (∃!xφ → (φ ↔ ∪{x∣φ} = x))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 955   ∈ wcel 957  ∃wex 979  ∃!weu 1379  {cab 1462  Vcvv 1808  {csn 2406  ∪cuni 2499

This theorem is referenced by:  reuuni1 2878

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-uni 2500

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