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Theorem vdwnnlem3 15539

Description: Lemma for vdwnn 15540. (Contributed by Mario Carneiro, 13-Sep-2014.) (Proof shortened by AV, 27-Sep-2020.)

**Hypotheses**
Ref	Expression
vdwnn.1	⊢ (𝜑 → 𝑅 ∈ Fin)
vdwnn.2	⊢ (𝜑 → 𝐹:ℕ⟶𝑅)
vdwnn.3	⊢ 𝑆 = {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐})}
vdwnn.4	⊢ (𝜑 → ∀𝑐 ∈ 𝑅 𝑆 ≠ ∅)

**Assertion**
Ref	Expression
vdwnnlem3	⊢ ¬ 𝜑

Distinct variable groups: 𝑎,𝑑,𝑘,𝑚,𝑐 𝜑,𝑎,𝑐,𝑑 𝑅,𝑎,𝑐,𝑑 𝐹,𝑎 𝑘,𝑐,𝐹,𝑑,𝑚 𝑆,𝑎,𝑑,𝑘,𝑚 Allowed substitution hints: 𝜑(𝑘,𝑚) 𝑅(𝑘,𝑚) 𝑆(𝑐)

Proof of Theorem vdwnnlem3 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vdwnn.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ Fin)
2		vdwnn.3	. . . . . . 7 ⊢ 𝑆 = {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐})}
3		ssrab2 3650	. . . . . . 7 ⊢ {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐})} ⊆ ℕ
4	2, 3	eqsstri 3598	. . . . . 6 ⊢ 𝑆 ⊆ ℕ
5		nnuz 11599	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
6	4, 5	sseqtri 3600	. . . . . . 7 ⊢ 𝑆 ⊆ (ℤ_≥‘1)
7		vdwnn.4	. . . . . . . 8 ⊢ (𝜑 → ∀𝑐 ∈ 𝑅 𝑆 ≠ ∅)
8	7	r19.21bi 2916	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑅) → 𝑆 ≠ ∅)
9		infssuzcl 11648	. . . . . . 7 ⊢ ((𝑆 ⊆ (ℤ_≥‘1) ∧ 𝑆 ≠ ∅) → inf(𝑆, ℝ, < ) ∈ 𝑆)
10	6, 8, 9	sylancr 694	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑅) → inf(𝑆, ℝ, < ) ∈ 𝑆)
11	4, 10	sseldi 3566	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑅) → inf(𝑆, ℝ, < ) ∈ ℕ)
12	11	nnred 10912	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑅) → inf(𝑆, ℝ, < ) ∈ ℝ)
13	12	ralrimiva 2949	. . 3 ⊢ (𝜑 → ∀𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ∈ ℝ)
14		fimaxre3 10849	. . 3 ⊢ ((𝑅 ∈ Fin ∧ ∀𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ≤ 𝑥)
15	1, 13, 14	syl2anc 691	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ≤ 𝑥)
16		vdwnn.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ⟶𝑅)
17		1nn 10908	. . . . . . . . 9 ⊢ 1 ∈ ℕ
18		ffvelrn 6265	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶𝑅 ∧ 1 ∈ ℕ) → (𝐹‘1) ∈ 𝑅)
19	16, 17, 18	sylancl 693	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘1) ∈ 𝑅)
20		ne0i 3880	. . . . . . . 8 ⊢ ((𝐹‘1) ∈ 𝑅 → 𝑅 ≠ ∅)
21	19, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ≠ ∅)
22	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑅 ≠ ∅)
23		r19.2z 4012	. . . . . . 7 ⊢ ((𝑅 ≠ ∅ ∧ ∀𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ≤ 𝑥) → ∃𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ≤ 𝑥)
24	23	ex 449	. . . . . 6 ⊢ (𝑅 ≠ ∅ → (∀𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ≤ 𝑥 → ∃𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ≤ 𝑥))
25	22, 24	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ≤ 𝑥 → ∃𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ≤ 𝑥))
26		simplr 788	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → 𝑥 ∈ ℝ)
27		fllep1 12464	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
28	26, 27	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → 𝑥 ≤ ((⌊‘𝑥) + 1))
29	12	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → inf(𝑆, ℝ, < ) ∈ ℝ)
30	26	flcld 12461	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → (⌊‘𝑥) ∈ ℤ)
31	30	peano2zd 11361	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → ((⌊‘𝑥) + 1) ∈ ℤ)
32	31	zred 11358	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → ((⌊‘𝑥) + 1) ∈ ℝ)
33		letr 10010	. . . . . . . . . 10 ⊢ ((inf(𝑆, ℝ, < ) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ ((⌊‘𝑥) + 1) ∈ ℝ) → ((inf(𝑆, ℝ, < ) ≤ 𝑥 ∧ 𝑥 ≤ ((⌊‘𝑥) + 1)) → inf(𝑆, ℝ, < ) ≤ ((⌊‘𝑥) + 1)))
34	29, 26, 32, 33	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → ((inf(𝑆, ℝ, < ) ≤ 𝑥 ∧ 𝑥 ≤ ((⌊‘𝑥) + 1)) → inf(𝑆, ℝ, < ) ≤ ((⌊‘𝑥) + 1)))
35	28, 34	mpan2d 706	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → (inf(𝑆, ℝ, < ) ≤ 𝑥 → inf(𝑆, ℝ, < ) ≤ ((⌊‘𝑥) + 1)))
36	11	adantlr 747	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → inf(𝑆, ℝ, < ) ∈ ℕ)
37	36	nnzd 11357	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → inf(𝑆, ℝ, < ) ∈ ℤ)
38		eluz 11577	. . . . . . . . . 10 ⊢ ((inf(𝑆, ℝ, < ) ∈ ℤ ∧ ((⌊‘𝑥) + 1) ∈ ℤ) → (((⌊‘𝑥) + 1) ∈ (ℤ_≥‘inf(𝑆, ℝ, < )) ↔ inf(𝑆, ℝ, < ) ≤ ((⌊‘𝑥) + 1)))
39	37, 31, 38	syl2anc 691	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → (((⌊‘𝑥) + 1) ∈ (ℤ_≥‘inf(𝑆, ℝ, < )) ↔ inf(𝑆, ℝ, < ) ≤ ((⌊‘𝑥) + 1)))
40		simpll 786	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → 𝜑)
41	10	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → inf(𝑆, ℝ, < ) ∈ 𝑆)
42	1, 16, 2	vdwnnlem2 15538	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘inf(𝑆, ℝ, < ))) → (inf(𝑆, ℝ, < ) ∈ 𝑆 → ((⌊‘𝑥) + 1) ∈ 𝑆))
43	42	impancom 455	. . . . . . . . . 10 ⊢ ((𝜑 ∧ inf(𝑆, ℝ, < ) ∈ 𝑆) → (((⌊‘𝑥) + 1) ∈ (ℤ_≥‘inf(𝑆, ℝ, < )) → ((⌊‘𝑥) + 1) ∈ 𝑆))
44	40, 41, 43	syl2anc 691	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → (((⌊‘𝑥) + 1) ∈ (ℤ_≥‘inf(𝑆, ℝ, < )) → ((⌊‘𝑥) + 1) ∈ 𝑆))
45	39, 44	sylbird 249	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → (inf(𝑆, ℝ, < ) ≤ ((⌊‘𝑥) + 1) → ((⌊‘𝑥) + 1) ∈ 𝑆))
46	35, 45	syld 46	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → (inf(𝑆, ℝ, < ) ≤ 𝑥 → ((⌊‘𝑥) + 1) ∈ 𝑆))
47	4	sseli 3564	. . . . . . . 8 ⊢ (((⌊‘𝑥) + 1) ∈ 𝑆 → ((⌊‘𝑥) + 1) ∈ ℕ)
48	47	nnnn0d 11228	. . . . . . 7 ⊢ (((⌊‘𝑥) + 1) ∈ 𝑆 → ((⌊‘𝑥) + 1) ∈ ℕ₀)
49	46, 48	syl6 34	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → (inf(𝑆, ℝ, < ) ≤ 𝑥 → ((⌊‘𝑥) + 1) ∈ ℕ₀))
50	49	rexlimdva 3013	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ≤ 𝑥 → ((⌊‘𝑥) + 1) ∈ ℕ₀))
51	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑥) + 1) ∈ ℕ₀) → 𝑅 ∈ Fin)
52	16	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑥) + 1) ∈ ℕ₀) → 𝐹:ℕ⟶𝑅)
53		simpr 476	. . . . . . . 8 ⊢ ((𝜑 ∧ ((⌊‘𝑥) + 1) ∈ ℕ₀) → ((⌊‘𝑥) + 1) ∈ ℕ₀)
54		vdwnnlem1 15537	. . . . . . . 8 ⊢ ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅 ∧ ((⌊‘𝑥) + 1) ∈ ℕ₀) → ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(((⌊‘𝑥) + 1) − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐}))
55	51, 52, 53, 54	syl3anc 1318	. . . . . . 7 ⊢ ((𝜑 ∧ ((⌊‘𝑥) + 1) ∈ ℕ₀) → ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(((⌊‘𝑥) + 1) − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐}))
56	55	ex 449	. . . . . 6 ⊢ (𝜑 → (((⌊‘𝑥) + 1) ∈ ℕ₀ → ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(((⌊‘𝑥) + 1) − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐})))
57	56	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (((⌊‘𝑥) + 1) ∈ ℕ₀ → ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(((⌊‘𝑥) + 1) − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐})))
58	25, 50, 57	3syld 58	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ≤ 𝑥 → ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(((⌊‘𝑥) + 1) − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐})))
59		oveq1 6556	. . . . . . . . . . . . 13 ⊢ (𝑘 = ((⌊‘𝑥) + 1) → (𝑘 − 1) = (((⌊‘𝑥) + 1) − 1))
60	59	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (𝑘 = ((⌊‘𝑥) + 1) → (0...(𝑘 − 1)) = (0...(((⌊‘𝑥) + 1) − 1)))
61	60	raleqdv 3121	. . . . . . . . . . 11 ⊢ (𝑘 = ((⌊‘𝑥) + 1) → (∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐}) ↔ ∀𝑚 ∈ (0...(((⌊‘𝑥) + 1) − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐})))
62	61	2rexbidv 3039	. . . . . . . . . 10 ⊢ (𝑘 = ((⌊‘𝑥) + 1) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(((⌊‘𝑥) + 1) − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐})))
63	62	notbid 307	. . . . . . . . 9 ⊢ (𝑘 = ((⌊‘𝑥) + 1) → (¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐}) ↔ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(((⌊‘𝑥) + 1) − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐})))
64	63, 2	elrab2 3333	. . . . . . . 8 ⊢ (((⌊‘𝑥) + 1) ∈ 𝑆 ↔ (((⌊‘𝑥) + 1) ∈ ℕ ∧ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(((⌊‘𝑥) + 1) − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐})))
65	64	simprbi 479	. . . . . . 7 ⊢ (((⌊‘𝑥) + 1) ∈ 𝑆 → ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(((⌊‘𝑥) + 1) − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐}))
66	46, 65	syl6 34	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑐 ∈ 𝑅) → (inf(𝑆, ℝ, < ) ≤ 𝑥 → ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(((⌊‘𝑥) + 1) − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐})))
67	66	ralimdva 2945	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ≤ 𝑥 → ∀𝑐 ∈ 𝑅 ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(((⌊‘𝑥) + 1) − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐})))
68		ralnex 2975	. . . . 5 ⊢ (∀𝑐 ∈ 𝑅 ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(((⌊‘𝑥) + 1) − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐}) ↔ ¬ ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(((⌊‘𝑥) + 1) − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐}))
69	67, 68	syl6ib 240	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ≤ 𝑥 → ¬ ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(((⌊‘𝑥) + 1) − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐹 “ {𝑐})))
70	58, 69	pm2.65d 186	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ¬ ∀𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ≤ 𝑥)
71	70	nrexdv 2984	. 2 ⊢ (𝜑 → ¬ ∃𝑥 ∈ ℝ ∀𝑐 ∈ 𝑅 inf(𝑆, ℝ, < ) ≤ 𝑥)
72	15, 71	pm2.65i 184	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 ⊆ wss 3540 ∅c0 3874 {csn 4125 class class class wbr 4583 ^◡ccnv 5037 “ cima 5041 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 infcinf 8230 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 ⌊cfl 12453

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fl 12455 df-hash 12980 df-vdwap 15510 df-vdwmc 15511 df-vdwpc 15512

This theorem is referenced by: vdwnn 15540

W3C validator