Theorem rexfiuz 11670

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 2740	. . . 4 ⊢ (𝑥 = ∅ → (∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ ∅ 𝜑))
2	1	rexralbidv 2568	. . 3 ⊢ (𝑥 = ∅ → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑))
3		raleq 2740	. . 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 2740	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ 𝑦 𝜑))
6	5	rexralbidv 2568	. . 3 ⊢ (𝑥 = 𝑦 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑))
7		raleq 2740	. . 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 2740	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑))
10	9	rexralbidv 2568	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑))
11		raleq 2740	. . 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 2740	. . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ 𝐴 𝜑))
14	13	rexralbidv 2568	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝐴 𝜑))
15		raleq 2740	. . 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 9587	. . . . 5 ⊢ 0 ∈ ℤ
18		elex2 2829	. . . . 5 ⊢ (0 ∈ ℤ → ∃𝑗 𝑗 ∈ ℤ)
19	17, 18	ax-mp 5	. . . 4 ⊢ ∃𝑗 𝑗 ∈ ℤ
20		ral0 3610	. . . . 5 ⊢ ∀𝑛 ∈ ∅ 𝜑
21	20	rgen2w 2598	. . . 4 ⊢ ∀𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑
22		r19.2m 3595	. . . 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 3610	. . 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 11669	. . . . 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 3399	. . . . . . 7 ⊢ (∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ (∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
29	28	ralbii 2548	. . . . . 6 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
30	29	rexbii 2549	. . . . 5 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
31		ralsnsg 3725	. . . . . . . 8 ⊢ (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ [𝑧 / 𝑛]∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))
32		sbcrex 3121	. . . . . . . . 9 ⊢ ([𝑧 / 𝑛]∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ [𝑧 / 𝑛]∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)
33		ralcom 2706	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ ∀𝑛 ∈ {𝑧}∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)
34		ralsnsg 3725	. . . . . . . . . . 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 2543	. . . . . . . . 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 2816	. . . . . 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 3399	. . . 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 7145	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 1398  ∃wex 1541   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521  Vcvv 2812  [wsbc 3041   ∪ cun 3208  ∅c0 3507  {csn 3688  ‘cfv 5351  Fincfn 6974  0cc0 8126  ℤcz 9576  ℤ_≥cuz 9852

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-er 6766  df-en 6975  df-fin 6977  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-inn 9237  df-n0 9496  df-z 9577  df-uz 9853

This theorem is referenced by: (None)