Theorem oprabid 5551

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 20-Mar-2013.)

Assertion

Ref	Expression
oprabid	⊢ (⟨⟨x, y⟩, z⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ φ)

Proof of Theorem oprabid

Dummy variables a r s t w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . 4 ⊢ x ∈ V
2		vex 2863	. . . 4 ⊢ y ∈ V
3	1, 2	opex 4589	. . 3 ⊢ ⟨x, y⟩ ∈ V
4		vex 2863	. . 3 ⊢ z ∈ V
5	3, 4	opex 4589	. 2 ⊢ ⟨⟨x, y⟩, z⟩ ∈ V
6	3, 4	eqvinop 4607	. . . . 5 ⊢ (w = ⟨⟨x, y⟩, z⟩ ↔ ∃a∃t(w = ⟨a, t⟩ ∧ ⟨a, t⟩ = ⟨⟨x, y⟩, z⟩))
7	6	biimpi 186	. . . 4 ⊢ (w = ⟨⟨x, y⟩, z⟩ → ∃a∃t(w = ⟨a, t⟩ ∧ ⟨a, t⟩ = ⟨⟨x, y⟩, z⟩))
8		eqeq1 2359	. . . . . . . 8 ⊢ (w = ⟨a, t⟩ → (w = ⟨⟨x, y⟩, z⟩ ↔ ⟨a, t⟩ = ⟨⟨x, y⟩, z⟩))
9		opth 4603	. . . . . . . . 9 ⊢ (⟨a, t⟩ = ⟨⟨x, y⟩, z⟩ ↔ (a = ⟨x, y⟩ ∧ t = z))
10	9	simplbi 446	. . . . . . . 8 ⊢ (⟨a, t⟩ = ⟨⟨x, y⟩, z⟩ → a = ⟨x, y⟩)
11	8, 10	syl6bi 219	. . . . . . 7 ⊢ (w = ⟨a, t⟩ → (w = ⟨⟨x, y⟩, z⟩ → a = ⟨x, y⟩))
12	1, 2	eqvinop 4607	. . . . . . . . 9 ⊢ (a = ⟨x, y⟩ ↔ ∃r∃s(a = ⟨r, s⟩ ∧ ⟨r, s⟩ = ⟨x, y⟩))
13		opeq1 4579	. . . . . . . . . . . . 13 ⊢ (a = ⟨r, s⟩ → ⟨a, t⟩ = ⟨⟨r, s⟩, t⟩)
14	13	eqeq2d 2364	. . . . . . . . . . . 12 ⊢ (a = ⟨r, s⟩ → (w = ⟨a, t⟩ ↔ w = ⟨⟨r, s⟩, t⟩))
15		opth 4603	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ ↔ (⟨x, y⟩ = ⟨r, s⟩ ∧ z = t))
16		opth 4603	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨x, y⟩ = ⟨r, s⟩ ↔ (x = r ∧ y = s))
17	16	anbi1i 676	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨x, y⟩ = ⟨r, s⟩ ∧ z = t) ↔ ((x = r ∧ y = s) ∧ z = t))
18	15, 17	bitri 240	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ ↔ ((x = r ∧ y = s) ∧ z = t))
19	18	anbi1i 676	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ ∧ φ) ↔ (((x = r ∧ y = s) ∧ z = t) ∧ φ))
20		anass 630	. . . . . . . . . . . . . . . . 17 ⊢ ((((x = r ∧ y = s) ∧ z = t) ∧ φ) ↔ ((x = r ∧ y = s) ∧ (z = t ∧ φ)))
21		anass 630	. . . . . . . . . . . . . . . . 17 ⊢ (((x = r ∧ y = s) ∧ (z = t ∧ φ)) ↔ (x = r ∧ (y = s ∧ (z = t ∧ φ))))
22	19, 20, 21	3bitri 262	. . . . . . . . . . . . . . . 16 ⊢ ((⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ ∧ φ) ↔ (x = r ∧ (y = s ∧ (z = t ∧ φ))))
23	22	3exbii 1584	. . . . . . . . . . . . . . 15 ⊢ (∃x∃y∃z(⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ ∧ φ) ↔ ∃x∃y∃z(x = r ∧ (y = s ∧ (z = t ∧ φ))))
24		nfcvf2 2513	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ ∀x x = z → Ⅎzx)
25		nfcvd 2491	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ ∀x x = z → Ⅎzr)
26	24, 25	nfeqd 2504	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ∀x x = z → Ⅎz x = r)
27	26	exdistrf 1971	. . . . . . . . . . . . . . . . . 18 ⊢ (∃x∃z(x = r ∧ (y = s ∧ (z = t ∧ φ))) → ∃x(x = r ∧ ∃z(y = s ∧ (z = t ∧ φ))))
28	27	eximi 1576	. . . . . . . . . . . . . . . . 17 ⊢ (∃y∃x∃z(x = r ∧ (y = s ∧ (z = t ∧ φ))) → ∃y∃x(x = r ∧ ∃z(y = s ∧ (z = t ∧ φ))))
29		excom 1741	. . . . . . . . . . . . . . . . 17 ⊢ (∃x∃y∃z(x = r ∧ (y = s ∧ (z = t ∧ φ))) ↔ ∃y∃x∃z(x = r ∧ (y = s ∧ (z = t ∧ φ))))
30		excom 1741	. . . . . . . . . . . . . . . . 17 ⊢ (∃x∃y(x = r ∧ ∃z(y = s ∧ (z = t ∧ φ))) ↔ ∃y∃x(x = r ∧ ∃z(y = s ∧ (z = t ∧ φ))))
31	28, 29, 30	3imtr4i 257	. . . . . . . . . . . . . . . 16 ⊢ (∃x∃y∃z(x = r ∧ (y = s ∧ (z = t ∧ φ))) → ∃x∃y(x = r ∧ ∃z(y = s ∧ (z = t ∧ φ))))
32		nfcvf2 2513	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ ∀x x = y → Ⅎyx)
33		nfcvd 2491	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ ∀x x = y → Ⅎyr)
34	32, 33	nfeqd 2504	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∀x x = y → Ⅎy x = r)
35	34	exdistrf 1971	. . . . . . . . . . . . . . . 16 ⊢ (∃x∃y(x = r ∧ ∃z(y = s ∧ (z = t ∧ φ))) → ∃x(x = r ∧ ∃y∃z(y = s ∧ (z = t ∧ φ))))
36		nfcvf2 2513	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ ∀y y = z → Ⅎzy)
37		nfcvd 2491	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ ∀y y = z → Ⅎzs)
38	36, 37	nfeqd 2504	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ∀y y = z → Ⅎz y = s)
39	38	exdistrf 1971	. . . . . . . . . . . . . . . . . 18 ⊢ (∃y∃z(y = s ∧ (z = t ∧ φ)) → ∃y(y = s ∧ ∃z(z = t ∧ φ)))
40	39	anim2i 552	. . . . . . . . . . . . . . . . 17 ⊢ ((x = r ∧ ∃y∃z(y = s ∧ (z = t ∧ φ))) → (x = r ∧ ∃y(y = s ∧ ∃z(z = t ∧ φ))))
41	40	eximi 1576	. . . . . . . . . . . . . . . 16 ⊢ (∃x(x = r ∧ ∃y∃z(y = s ∧ (z = t ∧ φ))) → ∃x(x = r ∧ ∃y(y = s ∧ ∃z(z = t ∧ φ))))
42	31, 35, 41	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (∃x∃y∃z(x = r ∧ (y = s ∧ (z = t ∧ φ))) → ∃x(x = r ∧ ∃y(y = s ∧ ∃z(z = t ∧ φ))))
43	23, 42	sylbi 187	. . . . . . . . . . . . . 14 ⊢ (∃x∃y∃z(⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ ∧ φ) → ∃x(x = r ∧ ∃y(y = s ∧ ∃z(z = t ∧ φ))))
44		df-3an 936	. . . . . . . . . . . . . . . . 17 ⊢ ((x = r ∧ y = s ∧ z = t) ↔ ((x = r ∧ y = s) ∧ z = t))
45	18, 44	bitr4i 243	. . . . . . . . . . . . . . . 16 ⊢ (⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ ↔ (x = r ∧ y = s ∧ z = t))
46		euequ1 2292	. . . . . . . . . . . . . . . . . . 19 ⊢ ∃!x x = r
47		eupick 2267	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∃!x x = r ∧ ∃x(x = r ∧ ∃y(y = s ∧ ∃z(z = t ∧ φ)))) → (x = r → ∃y(y = s ∧ ∃z(z = t ∧ φ))))
48	46, 47	mpan 651	. . . . . . . . . . . . . . . . . 18 ⊢ (∃x(x = r ∧ ∃y(y = s ∧ ∃z(z = t ∧ φ))) → (x = r → ∃y(y = s ∧ ∃z(z = t ∧ φ))))
49		euequ1 2292	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∃!y y = s
50		eupick 2267	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃!y y = s ∧ ∃y(y = s ∧ ∃z(z = t ∧ φ))) → (y = s → ∃z(z = t ∧ φ)))
51	49, 50	mpan 651	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃y(y = s ∧ ∃z(z = t ∧ φ)) → (y = s → ∃z(z = t ∧ φ)))
52		euequ1 2292	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∃!z z = t
53		eupick 2267	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃!z z = t ∧ ∃z(z = t ∧ φ)) → (z = t → φ))
54	52, 53	mpan 651	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃z(z = t ∧ φ) → (z = t → φ))
55	51, 54	syl6 29	. . . . . . . . . . . . . . . . . 18 ⊢ (∃y(y = s ∧ ∃z(z = t ∧ φ)) → (y = s → (z = t → φ)))
56	48, 55	syl6 29	. . . . . . . . . . . . . . . . 17 ⊢ (∃x(x = r ∧ ∃y(y = s ∧ ∃z(z = t ∧ φ))) → (x = r → (y = s → (z = t → φ))))
57	56	3impd 1165	. . . . . . . . . . . . . . . 16 ⊢ (∃x(x = r ∧ ∃y(y = s ∧ ∃z(z = t ∧ φ))) → ((x = r ∧ y = s ∧ z = t) → φ))
58	45, 57	syl5bi 208	. . . . . . . . . . . . . . 15 ⊢ (∃x(x = r ∧ ∃y(y = s ∧ ∃z(z = t ∧ φ))) → (⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ → φ))
59	58	com12 27	. . . . . . . . . . . . . 14 ⊢ (⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ → (∃x(x = r ∧ ∃y(y = s ∧ ∃z(z = t ∧ φ))) → φ))
60	43, 59	syl5 28	. . . . . . . . . . . . 13 ⊢ (⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ → (∃x∃y∃z(⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ ∧ φ) → φ))
61		eqeq1 2359	. . . . . . . . . . . . . . 15 ⊢ (w = ⟨⟨r, s⟩, t⟩ → (w = ⟨⟨x, y⟩, z⟩ ↔ ⟨⟨r, s⟩, t⟩ = ⟨⟨x, y⟩, z⟩))
62		eqcom 2355	. . . . . . . . . . . . . . 15 ⊢ (⟨⟨r, s⟩, t⟩ = ⟨⟨x, y⟩, z⟩ ↔ ⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩)
63	61, 62	syl6bb 252	. . . . . . . . . . . . . 14 ⊢ (w = ⟨⟨r, s⟩, t⟩ → (w = ⟨⟨x, y⟩, z⟩ ↔ ⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩))
64	63	anbi1d 685	. . . . . . . . . . . . . . . 16 ⊢ (w = ⟨⟨r, s⟩, t⟩ → ((w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ (⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ ∧ φ)))
65	64	3exbidv 1629	. . . . . . . . . . . . . . 15 ⊢ (w = ⟨⟨r, s⟩, t⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ∃x∃y∃z(⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ ∧ φ)))
66	65	imbi1d 308	. . . . . . . . . . . . . 14 ⊢ (w = ⟨⟨r, s⟩, t⟩ → ((∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) → φ) ↔ (∃x∃y∃z(⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ ∧ φ) → φ)))
67	63, 66	imbi12d 311	. . . . . . . . . . . . 13 ⊢ (w = ⟨⟨r, s⟩, t⟩ → ((w = ⟨⟨x, y⟩, z⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) → φ)) ↔ (⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ → (∃x∃y∃z(⟨⟨x, y⟩, z⟩ = ⟨⟨r, s⟩, t⟩ ∧ φ) → φ))))
68	60, 67	mpbiri 224	. . . . . . . . . . . 12 ⊢ (w = ⟨⟨r, s⟩, t⟩ → (w = ⟨⟨x, y⟩, z⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) → φ)))
69	14, 68	syl6bi 219	. . . . . . . . . . 11 ⊢ (a = ⟨r, s⟩ → (w = ⟨a, t⟩ → (w = ⟨⟨x, y⟩, z⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) → φ))))
70	69	adantr 451	. . . . . . . . . 10 ⊢ ((a = ⟨r, s⟩ ∧ ⟨r, s⟩ = ⟨x, y⟩) → (w = ⟨a, t⟩ → (w = ⟨⟨x, y⟩, z⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) → φ))))
71	70	exlimivv 1635	. . . . . . . . 9 ⊢ (∃r∃s(a = ⟨r, s⟩ ∧ ⟨r, s⟩ = ⟨x, y⟩) → (w = ⟨a, t⟩ → (w = ⟨⟨x, y⟩, z⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) → φ))))
72	12, 71	sylbi 187	. . . . . . . 8 ⊢ (a = ⟨x, y⟩ → (w = ⟨a, t⟩ → (w = ⟨⟨x, y⟩, z⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) → φ))))
73	72	com3l 75	. . . . . . 7 ⊢ (w = ⟨a, t⟩ → (w = ⟨⟨x, y⟩, z⟩ → (a = ⟨x, y⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) → φ))))
74	11, 73	mpdd 36	. . . . . 6 ⊢ (w = ⟨a, t⟩ → (w = ⟨⟨x, y⟩, z⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) → φ)))
75	74	adantr 451	. . . . 5 ⊢ ((w = ⟨a, t⟩ ∧ ⟨a, t⟩ = ⟨⟨x, y⟩, z⟩) → (w = ⟨⟨x, y⟩, z⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) → φ)))
76	75	exlimivv 1635	. . . 4 ⊢ (∃a∃t(w = ⟨a, t⟩ ∧ ⟨a, t⟩ = ⟨⟨x, y⟩, z⟩) → (w = ⟨⟨x, y⟩, z⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) → φ)))
77	7, 76	mpcom 32	. . 3 ⊢ (w = ⟨⟨x, y⟩, z⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) → φ))
78		19.8a 1756	. . . . 5 ⊢ ((w = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ))
79		19.8a 1756	. . . . 5 ⊢ (∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ))
80		19.8a 1756	. . . . 5 ⊢ (∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ))
81	78, 79, 80	3syl 18	. . . 4 ⊢ ((w = ⟨⟨x, y⟩, z⟩ ∧ φ) → ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ))
82	81	ex 423	. . 3 ⊢ (w = ⟨⟨x, y⟩, z⟩ → (φ → ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)))
83	77, 82	impbid 183	. 2 ⊢ (w = ⟨⟨x, y⟩, z⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ φ))
84		df-oprab 5529	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {w ∣ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)}
85	5, 83, 84	elab2 2989	1 ⊢ (⟨⟨x, y⟩, z⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ φ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ⟨cop 4562  {coprab 5528

This proof depends on 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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-oprab 5529

This theorem is used by:  ovidig  5594