Theorem rexanuz2nf 45503

Description: A simple counterexample related to theorem rexanuz2 15388, demonstrating the necessity of its disjoint variable constraints. Here, 𝑗 appears free in 𝜑, showing that without these constraints, rexanuz2 15388 and similar theorems would not hold (see rexanre 15385 and rexanuz 15384). (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 12541	. . . . . . . 8 ⊢ 0 ∈ ℕ0
2		nn0ge0 12551	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → 0 ≤ 𝑘)
3	2	rgen 3063	. . . . . . . 8 ⊢ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘
4		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → (ℤ≥‘𝑗) = (ℤ≥‘0))
5		nn0uz 12920	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
6	4, 5	eqtr4di 2795	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → (ℤ≥‘𝑗) = ℕ0)
7	6	raleqdv 3326	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ ∀𝑘 ∈ ℕ0 (𝑗 = 0 ∧ 𝑗 ≤ 𝑘)))
8	2	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ (𝑗 = 0 ∧ 𝑗 ≤ 𝑘)) → 0 ≤ 𝑘)
9		simpll 767	. . . . . . . . . . . . 13 ⊢ (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 𝑗 = 0)
10		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 0 ≤ 𝑘)
11	9, 10	eqbrtrd 5165	. . . . . . . . . . . . 13 ⊢ (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 𝑗 ≤ 𝑘)
12	9, 11	jca 511	. . . . . . . . . . . 12 ⊢ (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → (𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
13	8, 12	impbida 801	. . . . . . . . . . 11 ⊢ ((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) → ((𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ 0 ≤ 𝑘))
14	13	ralbidva 3176	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (∀𝑘 ∈ ℕ0 (𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘))
15	7, 14	bitrd 279	. . . . . . . . 9 ⊢ (𝑗 = 0 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘))
16	15	rspcev 3622	. . . . . . . 8 ⊢ ((0 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘) → ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
17	1, 3, 16	mp2an 692	. . . . . . 7 ⊢ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘)
18		rexanuz2nf.1	. . . . . . . . 9 ⊢ 𝑍 = ℕ0
19		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑗ℕ0
20	18, 19	nfcxfr 2903	. . . . . . . 8 ⊢ Ⅎ𝑗𝑍
21	20, 19, 18	rexeqif 45171	. . . . . . 7 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
22	17, 21	mpbir 231	. . . . . 6 ⊢ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘)
23		rexanuz2nf.2	. . . . . . . 8 ⊢ (𝜑 ↔ (𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
24	23	ralbii 3093	. . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
25	24	rexbii 3094	. . . . . 6 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑗 = 0 ∧ 𝑗 ≤ 𝑘))
26	22, 25	mpbir 231	. . . . 5 ⊢ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑
27		1nn0 12542	. . . . . . . 8 ⊢ 1 ∈ ℕ0
28		nngt0 12297	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 0 < 𝑘)
29	28	rgen 3063	. . . . . . . 8 ⊢ ∀𝑘 ∈ ℕ 0 < 𝑘
30		fveq2 6906	. . . . . . . . . . 11 ⊢ (𝑗 = 1 → (ℤ≥‘𝑗) = (ℤ≥‘1))
31		nnuz 12921	. . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1)
32	30, 31	eqtr4di 2795	. . . . . . . . . 10 ⊢ (𝑗 = 1 → (ℤ≥‘𝑗) = ℕ)
33	32	raleqdv 3326	. . . . . . . . 9 ⊢ (𝑗 = 1 → (∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘 ↔ ∀𝑘 ∈ ℕ 0 < 𝑘))
34	33	rspcev 3622	. . . . . . . 8 ⊢ ((1 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ 0 < 𝑘) → ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘)
35	27, 29, 34	mp2an 692	. . . . . . 7 ⊢ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘
36	20, 19, 18	rexeqif 45171	. . . . . . 7 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘 ↔ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘)
37	35, 36	mpbir 231	. . . . . 6 ⊢ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘
38		rexanuz2nf.3	. . . . . . . 8 ⊢ (𝜓 ↔ 0 < 𝑘)
39	38	ralbii 3093	. . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)𝜓 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘)
40	39	rexbii 3094	. . . . . 6 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)0 < 𝑘)
41	37, 40	mpbir 231	. . . . 5 ⊢ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓
42	26, 41	pm3.2i 470	. . . 4 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓)
43		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑘 ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗)
44		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑗
45		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑘(ℤ≥‘𝑗)
46	5	uzid3 45446	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ0 → 𝑗 ∈ (ℤ≥‘𝑗))
47	46	adantr 480	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ0 ∧ 𝑗 = 0) → 𝑗 ∈ (ℤ≥‘𝑗))
48		0re 11263	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
49	48	ltnri 11370	. . . . . . . . . . . . 13 ⊢ ¬ 0 < 0
50	49	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → ¬ 0 < 0)
51		eqcom 2744	. . . . . . . . . . . . 13 ⊢ (𝑗 = 0 ↔ 0 = 𝑗)
52	51	biimpi 216	. . . . . . . . . . . 12 ⊢ (𝑗 = 0 → 0 = 𝑗)
53	50, 52	brneqtrd 45081	. . . . . . . . . . 11 ⊢ (𝑗 = 0 → ¬ 0 < 𝑗)
54	53	intnand 488	. . . . . . . . . 10 ⊢ (𝑗 = 0 → ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗))
55	54	adantl 481	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ0 ∧ 𝑗 = 0) → ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗))
56		breq2 5147	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑗 ≤ 𝑘 ↔ 𝑗 ≤ 𝑗))
57	56	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑗 = 0 ∧ 𝑗 ≤ 𝑘) ↔ (𝑗 = 0 ∧ 𝑗 ≤ 𝑗)))
58	23, 57	bitrid 283	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝜑 ↔ (𝑗 = 0 ∧ 𝑗 ≤ 𝑗)))
59		breq2 5147	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (0 < 𝑘 ↔ 0 < 𝑗))
60	38, 59	bitrid 283	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝜓 ↔ 0 < 𝑗))
61	58, 60	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝜓) ↔ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗)))
62	61	notbid 318	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (¬ (𝜑 ∧ 𝜓) ↔ ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗)))
63	43, 44, 45, 47, 55, 62	rspced 45172	. . . . . . . 8 ⊢ ((𝑗 ∈ ℕ0 ∧ 𝑗 = 0) → ∃𝑘 ∈ (ℤ≥‘𝑗) ¬ (𝜑 ∧ 𝜓))
64	46	adantr 480	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → 𝑗 ∈ (ℤ≥‘𝑗))
65		id 22	. . . . . . . . . . . 12 ⊢ (¬ 𝑗 = 0 → ¬ 𝑗 = 0)
66	65	intnanrd 489	. . . . . . . . . . 11 ⊢ (¬ 𝑗 = 0 → ¬ (𝑗 = 0 ∧ 𝑗 ≤ 𝑗))
67	66	intnanrd 489	. . . . . . . . . 10 ⊢ (¬ 𝑗 = 0 → ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗))
68	67	adantl 481	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → ¬ ((𝑗 = 0 ∧ 𝑗 ≤ 𝑗) ∧ 0 < 𝑗))
69	43, 44, 45, 64, 68, 62	rspced 45172	. . . . . . . 8 ⊢ ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → ∃𝑘 ∈ (ℤ≥‘𝑗) ¬ (𝜑 ∧ 𝜓))
70	63, 69	pm2.61dan 813	. . . . . . 7 ⊢ (𝑗 ∈ ℕ0 → ∃𝑘 ∈ (ℤ≥‘𝑗) ¬ (𝜑 ∧ 𝜓))
71		rexnal 3100	. . . . . . 7 ⊢ (∃𝑘 ∈ (ℤ≥‘𝑗) ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
72	70, 71	sylib 218	. . . . . 6 ⊢ (𝑗 ∈ ℕ0 → ¬ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
73	72	nrex 3074	. . . . 5 ⊢ ¬ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓)
74	20, 19, 18	rexeqif 45171	. . . . 5 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
75	73, 74	mtbir 323	. . . 4 ⊢ ¬ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓)
76	42, 75	pm3.2i 470	. . 3 ⊢ ((∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) ∧ ¬ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
77		annim 403	. . 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 230	. 2 ⊢ ¬ ((∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓))
79	78	nimnbi2 45169	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   class class class wbr 5143  ‘cfv 6561  0cc0 11155  1c1 11156   < clt 11295   ≤ cle 11296  ℕcn 12266  ℕ₀cn0 12526  ℤ_≥cuz 12878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879

This theorem is referenced by: (None)