Theorem rexanuz2nf 46097

Description: A simple counterexample related to theorem rexanuz2 15400, demonstrating the necessity of its disjoint variable constraints. Here, 𝑗 appears free in 𝜑, showing that without these constraints, rexanuz2 15400 and similar theorems would not hold (see rexanre 15397 and rexanuz 15396). (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 12518	. . . . . . . 8 ⊢ 0 ∈ ℕ0
2		nn0ge0 12528	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → 0 ≤ 𝑘)
3	2	rgen 3087	. . . . . . . 8 ⊢ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘
4		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → (ℤ≥‘𝑗) = (ℤ≥‘0))
5		nn0uz 12899	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
6	4, 5	eqtr4di 2822	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → (ℤ≥‘𝑗) = ℕ0)
7	6	raleqdv 3329	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ ∀𝑘 ∈ ℕ0 (𝑗 = 0 ∧ 𝑗 ≤ 𝑘)))
8	2	ad2antlr 739	. . . . . . . . . . . 12 ⊢ (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ (𝑗 = 0 ∧ 𝑗 ≤ 𝑘)) → 0 ≤ 𝑘)
9		simpll 778	. . . . . . . . . . . . 13 ⊢ (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 𝑗 = 0)
10		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 0 ≤ 𝑘)
11	9, 10	eqbrtrd 5137	. . . . . . . . . . . . 13 ⊢ (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 𝑗 ≤ 𝑘)
12	9, 11	jca 520	. . . . . . . . . . . 12 ⊢ (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → (𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
13	8, 12	impbida 812	. . . . . . . . . . 11 ⊢ ((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) → ((𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ 0 ≤ 𝑘))
14	13	ralbidva 3192	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (∀𝑘 ∈ ℕ0 (𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘))
15	7, 14	bitrd 282	. . . . . . . . 9 ⊢ (𝑗 = 0 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘))
16	15	rspcev 3590	. . . . . . . 8 ⊢ ((0 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘) → ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
17	1, 3, 16	mp2an 704	. . . . . . 7 ⊢ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘)
18		rexanuz2nf.1	. . . . . . . . 9 ⊢ 𝑍 = ℕ0
19		nfcv 2931	. . . . . . . . 9 ⊢ Ⅎ𝑗ℕ0
20	18, 19	nfcxfr 2929	. . . . . . . 8 ⊢ Ⅎ𝑗𝑍
21	20, 19, 18	rexeqif 45775	. . . . . . 7 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
22	17, 21	mpbir 234	. . . . . 6 ⊢ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘)
23		rexanuz2nf.2	. . . . . . . 8 ⊢ (𝜑 ↔ (𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
24	23	ralbii 3117	. . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
25	24	rexbii 3118	. . . . . 6 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
26	22, 25	mpbir 234	. . . . 5 ⊢ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑
27		1nn0 12519	. . . . . . . 8 ⊢ 1 ∈ ℕ0
28		nngt0 12266	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 0 < 𝑘)
29	28	rgen 3087	. . . . . . . 8 ⊢ ∀𝑘 ∈ ℕ 0 < 𝑘
30		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑗 = 1 → (ℤ≥‘𝑗) = (ℤ≥‘1))
31		nnuz 12900	. . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1)
32	30, 31	eqtr4di 2822	. . . . . . . . . 10 ⊢ (𝑗 = 1 → (ℤ≥‘𝑗) = ℕ)
33	32	raleqdv 3329	. . . . . . . . 9 ⊢ (𝑗 = 1 → (∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘 ↔ ∀𝑘 ∈ ℕ 0 < 𝑘))
34	33	rspcev 3590	. . . . . . . 8 ⊢ ((1 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ 0 < 𝑘) → ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘)
35	27, 29, 34	mp2an 704	. . . . . . 7 ⊢ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘
36	20, 19, 18	rexeqif 45775	. . . . . . 7 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘 ↔ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘)
37	35, 36	mpbir 234	. . . . . 6 ⊢ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘
38		rexanuz2nf.3	. . . . . . . 8 ⊢ (𝜓 ↔ 0 < 𝑘)
39	38	ralbii 3117	. . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)𝜓 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘)
40	39	rexbii 3118	. . . . . 6 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘)
41	37, 40	mpbir 234	. . . . 5 ⊢ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓
42	26, 41	pm3.2i 475	. . . 4 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓)
43		nfv 1941	. . . . . . . . 9 ⊢ Ⅎ𝑘 ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗)
44		nfcv 2931	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑗
45		nfcv 2931	. . . . . . . . 9 ⊢ Ⅎ𝑘(ℤ≥‘𝑗)
46	5	uzid3 46040	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ0 → 𝑗 ∈ (ℤ≥‘𝑗))
47	46	adantr 485	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ0 ∧ 𝑗 = 0) → 𝑗 ∈ (ℤ≥‘𝑗))
48		0re 11209	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
49	48	ltnri 11318	. . . . . . . . . . . . 13 ⊢ ¬ 0 < 0
50	49	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → ¬ 0 < 0)
51		eqcom 2776	. . . . . . . . . . . . 13 ⊢ (𝑗 = 0 ↔ 0 = 𝑗)
52	51	biimpi 219	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → 0 = 𝑗)
53	50, 52	brneqtrd 45687	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → ¬ 0 < 𝑗)
54	53	intnand 493	. . . . . . . . . 10 ⊢ (𝑗 = 0 → ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗))
55	54	adantl 486	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ0 ∧ 𝑗 = 0) → ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗))
56		breq2 5117	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑗 ≤ 𝑘 ↔ 𝑗 ≤ 𝑗))
57	56	anbi2d 641	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ (𝑗 = 0 ∧ 𝑗 ≤ 𝑗)))
58	23, 57	bitrid 286	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝜑 ↔ (𝑗 = 0 ∧ 𝑗 ≤ 𝑗)))
59		breq2 5117	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (0 < 𝑘 ↔ 0 < 𝑗))
60	38, 59	bitrid 286	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝜓 ↔ 0 < 𝑗))
61	58, 60	anbi12d 643	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝜓) ↔ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗)))
62	61	notbid 321	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (¬ (𝜑 ∧ 𝜓) ↔ ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗)))
63	43, 44, 45, 47, 55, 62	rspced 45776	. . . . . . . 8 ⊢ ((𝑗 ∈ ℕ0 ∧ 𝑗 = 0) → ∃𝑘 ∈ (ℤ≥‘𝑗) ¬ (𝜑 ∧ 𝜓))
64	46	adantr 485	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → 𝑗 ∈ (ℤ≥‘𝑗))
65		id 23	. . . . . . . . . . . 12 ⊢ (¬ 𝑗 = 0 → ¬ 𝑗 = 0)
66	65	intnanrd 494	. . . . . . . . . . 11 ⊢ (¬ 𝑗 = 0 → ¬ (𝑗 = 0 ∧ 𝑗 ≤ 𝑗))
67	66	intnanrd 494	. . . . . . . . . 10 ⊢ (¬ 𝑗 = 0 → ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗))
68	67	adantl 486	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗))
69	43, 44, 45, 64, 68, 62	rspced 45776	. . . . . . . 8 ⊢ ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → ∃𝑘 ∈ (ℤ≥‘𝑗) ¬ (𝜑 ∧ 𝜓))
70	63, 69	pm2.61dan 824	. . . . . . 7 ⊢ (𝑗 ∈ ℕ0 → ∃𝑘 ∈ (ℤ≥‘𝑗) ¬ (𝜑 ∧ 𝜓))
71		rexnal 3123	. . . . . . 7 ⊢ (∃𝑘 ∈ (ℤ≥‘𝑗) ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
72	70, 71	sylib 221	. . . . . 6 ⊢ (𝑗 ∈ ℕ0 → ¬ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
73	72	nrex 3099	. . . . 5 ⊢ ¬ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓)
74	20, 19, 18	rexeqif 45775	. . . . 5 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
75	73, 74	mtbir 326	. . . 4 ⊢ ¬ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓)
76	42, 75	pm3.2i 475	. . 3 ⊢ ((∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) ∧ ¬ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
77		annim 408	. . 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 233	. 2 ⊢ ¬ ((∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
79	78	nimnbi2 45773	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 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   class class class wbr 5113  ‘cfv 6537  0cc0 11099  1c1 11100   < clt 11242   ≤ cle 11243  ℕcn 12232  ℕ₀cn0 12503  ℤ_≥cuz 12861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-n0 12504  df-z 12591  df-uz 12862

This theorem is referenced by: (None)