Theorem moop2 1910

Description: "At most one" property of an ordered pair. The proof is a little tricky because we do not place any restrictions on class B.

Assertion

Ref	Expression
moop2	⊢ ∃*x A = ⟨B, x⟩

Distinct variable group(s):   x,A

Proof of Theorem moop2

Step	Hyp	Ref	Expression
1		cleqtr 1118	. . . . 5 ⊢ ((⟨B, x⟩ = A ∧ A = ⟨{z∣[y / x]z ∈ B}, y⟩) → ⟨B, x⟩ = ⟨{z∣[y / x]z ∈ B}, y⟩)
2		cleqcom 1103	. . . . 5 ⊢ (A = ⟨B, x⟩ ↔ ⟨B, x⟩ = A)
3	1, 2	sylanb 344	. . . 4 ⊢ ((A = ⟨B, x⟩ ∧ A = ⟨{z∣[y / x]z ∈ B}, y⟩) → ⟨B, x⟩ = ⟨{z∣[y / x]z ∈ B}, y⟩)
4		visset 1350	. . . . 5 ⊢ x ∈ V
5		visset 1350	. . . . 5 ⊢ y ∈ V
6	4, 5	opth2 1909	. . . 4 ⊢ (⟨B, x⟩ = ⟨{z∣[y / x]z ∈ B}, y⟩ → x = y)
7	3, 6	syl 12	. . 3 ⊢ ((A = ⟨B, x⟩ ∧ A = ⟨{z∣[y / x]z ∈ B}, y⟩) → x = y)
8	7	gen2 681	. 2 ⊢ ∀x∀y((A = ⟨B, x⟩ ∧ A = ⟨{z∣[y / x]z ∈ B}, y⟩) → x = y)
9		ax-17 925	. . . 4 ⊢ (w ∈ A → ∀x w ∈ A)
10		hbs1 986	. . . . . 6 ⊢ ([y / x]z ∈ B → ∀x[y / x]z ∈ B)
11	10	hbab 1096	. . . . 5 ⊢ (w ∈ {z∣[y / x]z ∈ B} → ∀x w ∈ {z∣[y / x]z ∈ B})
12		ax-17 925	. . . . 5 ⊢ (w ∈ y → ∀x w ∈ y)
13	11, 12	hbop 1879	. . . 4 ⊢ (w ∈ ⟨{z∣[y / x]z ∈ B}, y⟩ → ∀x w ∈ ⟨{z∣[y / x]z ∈ B}, y⟩)
14	9, 13	hbeq 1171	. . 3 ⊢ (A = ⟨{z∣[y / x]z ∈ B}, y⟩ → ∀x A = ⟨{z∣[y / x]z ∈ B}, y⟩)
15		sbab 1188	. . . . . 6 ⊢ (x = y → B = {z∣[y / x]z ∈ B})
16		opeq1 1876	. . . . . 6 ⊢ (B = {z∣[y / x]z ∈ B} → ⟨B, x⟩ = ⟨{z∣[y / x]z ∈ B}, x⟩)
17	15, 16	syl 12	. . . . 5 ⊢ (x = y → ⟨B, x⟩ = ⟨{z∣[y / x]z ∈ B}, x⟩)
18		opeq2 1877	. . . . 5 ⊢ (x = y → ⟨{z∣[y / x]z ∈ B}, x⟩ = ⟨{z∣[y / x]z ∈ B}, y⟩)
19	17, 18	eqtrd 1128	. . . 4 ⊢ (x = y → ⟨B, x⟩ = ⟨{z∣[y / x]z ∈ B}, y⟩)
20	19	cleq2d 1112	. . 3 ⊢ (x = y → (A = ⟨B, x⟩ ↔ A = ⟨{z∣[y / x]z ∈ B}, y⟩))
21	14, 20	mo4f 1028	. 2 ⊢ (∃*x A = ⟨B, x⟩ ↔ ∀x∀y((A = ⟨B, x⟩ ∧ A = ⟨{z∣[y / x]z ∈ B}, y⟩) → x = y))
22	8, 21	mpbir 165	1 ⊢ ∃*x A = ⟨B, x⟩

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672   = weq 797   ∈ wel 803  [wsb 852  ∃*wmo 1008  {cab 1090   = wceq 1091   ∈ wcel 1092  ⟨cop 1810

This theorem is referenced by:  euop2 1912

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  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-op 1815

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