Theorem rexfiuz 11350

Description: Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.)

Assertion

Ref	Expression
rexfiuz	⊢ (𝐴 ∈ Fin → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝐴 𝜑 ↔ ∀𝑛 ∈ 𝐴 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))

Distinct variable groups:   𝑗,𝑘,𝑛,𝐴   𝜑,𝑗

Allowed substitution hints:   𝜑(𝑘,𝑛)

Proof of Theorem rexfiuz

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 2703	. . . 4 ⊢ (𝑥 = ∅ → (∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ ∅ 𝜑))
2	1	rexralbidv 2533	. . 3 ⊢ (𝑥 = ∅ → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑))
3		raleq 2703	. . 3 ⊢ (𝑥 = ∅ → (∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
4	2, 3	bibi12d 235	. 2 ⊢ (𝑥 = ∅ → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)))
5		raleq 2703	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ 𝑦 𝜑))
6	5	rexralbidv 2533	. . 3 ⊢ (𝑥 = 𝑦 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑))
7		raleq 2703	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∀𝑛 ∈ 𝑦 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
8	6, 7	bibi12d 235	. 2 ⊢ (𝑥 = 𝑦 → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ↔ ∀𝑛 ∈ 𝑦 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)))
9		raleq 2703	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑))
10	9	rexralbidv 2533	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑))
11		raleq 2703	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
12	10, 11	bibi12d 235	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)))
13		raleq 2703	. . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ 𝐴 𝜑))
14	13	rexralbidv 2533	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝐴 𝜑))
15		raleq 2703	. . 3 ⊢ (𝑥 = 𝐴 → (∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∀𝑛 ∈ 𝐴 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
16	14, 15	bibi12d 235	. 2 ⊢ (𝑥 = 𝐴 → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝐴 𝜑 ↔ ∀𝑛 ∈ 𝐴 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)))
17		0z 9396	. . . . 5 ⊢ 0 ∈ ℤ
18		elex2 2790	. . . . 5 ⊢ (0 ∈ ℤ → ∃𝑗 𝑗 ∈ ℤ)
19	17, 18	ax-mp 5	. . . 4 ⊢ ∃𝑗 𝑗 ∈ ℤ
20		ral0 3564	. . . . 5 ⊢ ∀𝑛 ∈ ∅ 𝜑
21	20	rgen2w 2563	. . . 4 ⊢ ∀𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑
22		r19.2m 3549	. . . 4 ⊢ ((∃𝑗 𝑗 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑)
23	19, 21, 22	mp2an 426	. . 3 ⊢ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑
24		ral0 3564	. . 3 ⊢ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑
25	23, 24	2th 174	. 2 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)
26		anbi1 466	. . . 4 ⊢ ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ↔ ∀𝑛 ∈ 𝑦 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) ↔ (∀𝑛 ∈ 𝑦 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)))
27		rexanuz 11349	. . . . 5 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑))
28		ralunb 3356	. . . . . . 7 ⊢ (∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ (∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
29	28	ralbii 2513	. . . . . 6 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
30	29	rexbii 2514	. . . . 5 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
31		ralsnsg 3672	. . . . . . . 8 ⊢ (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ [𝑧 / 𝑛]∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
32		sbcrex 3080	. . . . . . . . 9 ⊢ ([𝑧 / 𝑛]∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ [𝑧 / 𝑛]∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)
33		ralcom 2670	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ ∀𝑛 ∈ {𝑧}∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)
34		ralsnsg 3672	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ [𝑧 / 𝑛]∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
35	33, 34	bitrid 192	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ [𝑧 / 𝑛]∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
36	35	rexbidv 2508	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ ∃𝑗 ∈ ℤ [𝑧 / 𝑛]∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
37	32, 36	bitr4id 199	. . . . . . . 8 ⊢ (𝑧 ∈ V → ([𝑧 / 𝑛]∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑))
38	31, 37	bitrd 188	. . . . . . 7 ⊢ (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑))
39	38	elv 2777	. . . . . 6 ⊢ (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑)
40	39	anbi2i 457	. . . . 5 ⊢ ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑))
41	27, 30, 40	3bitr4i 212	. . . 4 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
42		ralunb 3356	. . . 4 ⊢ (∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ (∀𝑛 ∈ 𝑦 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
43	26, 41, 42	3bitr4g 223	. . 3 ⊢ ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ↔ ∀𝑛 ∈ 𝑦 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
44	43	a1i 9	. 2 ⊢ (𝑦 ∈ Fin → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ↔ ∀𝑛 ∈ 𝑦 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)))
45	4, 8, 12, 16, 25, 44	findcard2 6998	1 ⊢ (𝐴 ∈ Fin → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝐴 𝜑 ↔ ∀𝑛 ∈ 𝐴 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  Vcvv 2773  [wsbc 3000   ∪ cun 3166  ∅c0 3462  {csn 3635  ‘cfv 5277  Fincfn 6837  0cc0 7938  ℤcz 9385  ℤ_≥cuz 9661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4164  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-iinf 4641  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-addcom 8038  ax-addass 8040  ax-distr 8042  ax-i2m1 8043  ax-0lt1 8044  ax-0id 8046  ax-rnegex 8047  ax-cnre 8049  ax-pre-ltirr 8050  ax-pre-ltwlin 8051  ax-pre-lttrn 8052  ax-pre-apti 8053  ax-pre-ltadd 8054

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-if 3574  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-tr 4148  df-id 4345  df-iord 4418  df-on 4420  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-er 6630  df-en 6838  df-fin 6840  df-pnf 8122  df-mnf 8123  df-xr 8124  df-ltxr 8125  df-le 8126  df-sub 8258  df-neg 8259  df-inn 9050  df-n0 9309  df-z 9386  df-uz 9662

This theorem is referenced by: (None)