Theorem rexanuz2nf 43974

Description: A simple counterexample related to theorem rexanuz2 15278, demonstrating the necessity of its disjoint variable constraints. Here, 𝑗 appears free in 𝜑, showing that without these constraints, rexanuz2 15278 and similar theorems would not hold (see rexanre 15275 and rexanuz 15274). (Contributed by Glauco Siliprandi, 15-Feb-2025.)

Hypotheses

Ref	Expression
rexanuz2nf.1	⊢ 𝑍 = ℕ0
rexanuz2nf.2	⊢ (𝜑 ↔ (𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
rexanuz2nf.3	⊢ (𝜓 ↔ 0 < 𝑘)

Assertion

Ref	Expression
rexanuz2nf	⊢ ¬ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))

Distinct variable group:   𝑗,𝑘

Allowed substitution hints:   𝜑(𝑗,𝑘)   𝜓(𝑗,𝑘)   𝑍(𝑗,𝑘)

Proof of Theorem rexanuz2nf

Step	Hyp	Ref	Expression
1		0nn0 12469	. . . . . . . 8 ⊢ 0 ∈ ℕ0
2		nn0ge0 12479	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → 0 ≤ 𝑘)
3	2	rgen 3062	. . . . . . . 8 ⊢ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘
4		fveq2 6878	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → (ℤ≥‘𝑗) = (ℤ≥‘0))
5		nn0uz 12846	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
6	4, 5	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → (ℤ≥‘𝑗) = ℕ0)
7	6	raleqdv 3324	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ ∀𝑘 ∈ ℕ0 (𝑗 = 0 ∧ 𝑗 ≤ 𝑘)))
8	2	ad2antlr 725	. . . . . . . . . . . 12 ⊢ (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ (𝑗 = 0 ∧ 𝑗 ≤ 𝑘)) → 0 ≤ 𝑘)
9		simpll 765	. . . . . . . . . . . . 13 ⊢ (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 𝑗 = 0)
10		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 0 ≤ 𝑘)
11	9, 10	eqbrtrd 5163	. . . . . . . . . . . . 13 ⊢ (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 𝑗 ≤ 𝑘)
12	9, 11	jca 512	. . . . . . . . . . . 12 ⊢ (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → (𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
13	8, 12	impbida 799	. . . . . . . . . . 11 ⊢ ((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) → ((𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ 0 ≤ 𝑘))
14	13	ralbidva 3174	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (∀𝑘 ∈ ℕ0 (𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘))
15	7, 14	bitrd 278	. . . . . . . . 9 ⊢ (𝑗 = 0 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘))
16	15	rspcev 3609	. . . . . . . 8 ⊢ ((0 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘) → ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
17	1, 3, 16	mp2an 690	. . . . . . 7 ⊢ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘)
18		rexanuz2nf.1	. . . . . . . . 9 ⊢ 𝑍 = ℕ0
19		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑗ℕ0
20	18, 19	nfcxfr 2900	. . . . . . . 8 ⊢ Ⅎ𝑗𝑍
21	20, 19, 18	rexeqif 43630	. . . . . . 7 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
22	17, 21	mpbir 230	. . . . . 6 ⊢ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘)
23		rexanuz2nf.2	. . . . . . . 8 ⊢ (𝜑 ↔ (𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
24	23	ralbii 3092	. . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
25	24	rexbii 3093	. . . . . 6 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
26	22, 25	mpbir 230	. . . . 5 ⊢ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑
27		1nn0 12470	. . . . . . . 8 ⊢ 1 ∈ ℕ0
28		nngt0 12225	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 0 < 𝑘)
29	28	rgen 3062	. . . . . . . 8 ⊢ ∀𝑘 ∈ ℕ 0 < 𝑘
30		fveq2 6878	. . . . . . . . . . 11 ⊢ (𝑗 = 1 → (ℤ≥‘𝑗) = (ℤ≥‘1))
31		nnuz 12847	. . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1)
32	30, 31	eqtr4di 2789	. . . . . . . . . 10 ⊢ (𝑗 = 1 → (ℤ≥‘𝑗) = ℕ)
33	32	raleqdv 3324	. . . . . . . . 9 ⊢ (𝑗 = 1 → (∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘 ↔ ∀𝑘 ∈ ℕ 0 < 𝑘))
34	33	rspcev 3609	. . . . . . . 8 ⊢ ((1 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ 0 < 𝑘) → ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘)
35	27, 29, 34	mp2an 690	. . . . . . 7 ⊢ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘
36	20, 19, 18	rexeqif 43630	. . . . . . 7 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘 ↔ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘)
37	35, 36	mpbir 230	. . . . . 6 ⊢ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘
38		rexanuz2nf.3	. . . . . . . 8 ⊢ (𝜓 ↔ 0 < 𝑘)
39	38	ralbii 3092	. . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)𝜓 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘)
40	39	rexbii 3093	. . . . . 6 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘)
41	37, 40	mpbir 230	. . . . 5 ⊢ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓
42	26, 41	pm3.2i 471	. . . 4 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓)
43		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑘 ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗)
44		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑗
45		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑘(ℤ≥‘𝑗)
46	5	uzid3 43916	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ0 → 𝑗 ∈ (ℤ≥‘𝑗))
47	46	adantr 481	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ0 ∧ 𝑗 = 0) → 𝑗 ∈ (ℤ≥‘𝑗))
48		0re 11198	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
49	48	ltnri 11305	. . . . . . . . . . . . 13 ⊢ ¬ 0 < 0
50	49	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → ¬ 0 < 0)
51		eqcom 2738	. . . . . . . . . . . . 13 ⊢ (𝑗 = 0 ↔ 0 = 𝑗)
52	51	biimpi 215	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → 0 = 𝑗)
53	50, 52	brneqtrd 43534	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → ¬ 0 < 𝑗)
54	53	intnand 489	. . . . . . . . . 10 ⊢ (𝑗 = 0 → ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗))
55	54	adantl 482	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ0 ∧ 𝑗 = 0) → ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗))
56		breq2 5145	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑗 ≤ 𝑘 ↔ 𝑗 ≤ 𝑗))
57	56	anbi2d 629	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ (𝑗 = 0 ∧ 𝑗 ≤ 𝑗)))
58	23, 57	bitrid 282	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝜑 ↔ (𝑗 = 0 ∧ 𝑗 ≤ 𝑗)))
59		breq2 5145	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (0 < 𝑘 ↔ 0 < 𝑗))
60	38, 59	bitrid 282	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝜓 ↔ 0 < 𝑗))
61	58, 60	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝜓) ↔ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗)))
62	61	notbid 317	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (¬ (𝜑 ∧ 𝜓) ↔ ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗)))
63	43, 44, 45, 47, 55, 62	rspced 43631	. . . . . . . 8 ⊢ ((𝑗 ∈ ℕ0 ∧ 𝑗 = 0) → ∃𝑘 ∈ (ℤ≥‘𝑗) ¬ (𝜑 ∧ 𝜓))
64	46	adantr 481	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → 𝑗 ∈ (ℤ≥‘𝑗))
65		id 22	. . . . . . . . . . . 12 ⊢ (¬ 𝑗 = 0 → ¬ 𝑗 = 0)
66	65	intnanrd 490	. . . . . . . . . . 11 ⊢ (¬ 𝑗 = 0 → ¬ (𝑗 = 0 ∧ 𝑗 ≤ 𝑗))
67	66	intnanrd 490	. . . . . . . . . 10 ⊢ (¬ 𝑗 = 0 → ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗))
68	67	adantl 482	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗))
69	43, 44, 45, 64, 68, 62	rspced 43631	. . . . . . . 8 ⊢ ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → ∃𝑘 ∈ (ℤ≥‘𝑗) ¬ (𝜑 ∧ 𝜓))
70	63, 69	pm2.61dan 811	. . . . . . 7 ⊢ (𝑗 ∈ ℕ0 → ∃𝑘 ∈ (ℤ≥‘𝑗) ¬ (𝜑 ∧ 𝜓))
71		rexnal 3099	. . . . . . 7 ⊢ (∃𝑘 ∈ (ℤ≥‘𝑗) ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
72	70, 71	sylib 217	. . . . . 6 ⊢ (𝑗 ∈ ℕ0 → ¬ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
73	72	nrex 3073	. . . . 5 ⊢ ¬ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓)
74	20, 19, 18	rexeqif 43630	. . . . 5 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
75	73, 74	mtbir 322	. . . 4 ⊢ ¬ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓)
76	42, 75	pm3.2i 471	. . 3 ⊢ ((∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) ∧ ¬ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
77		annim 404	. . 3 ⊢ (((∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) ∧ ¬ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓)) ↔ ¬ ((∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓)))
78	76, 77	mpbi 229	. 2 ⊢ ¬ ((∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
79	78	nimnbi2 43628	1 ⊢ ¬ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069   class class class wbr 5141  ‘cfv 6532  0cc0 11092  1c1 11093   < clt 11230   ≤ cle 11231  ℕcn 12194  ℕ₀cn0 12454  ℤ_≥cuz 12804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-n0 12455  df-z 12541  df-uz 12805

This theorem is referenced by: (None)