Theorem swrdnd0 14811

Description: The value of a subword operation for inproper arguments is the empty set. (Contributed by AV, 2-Dec-2022.)

Assertion

Ref	Expression
swrdnd0	⊢ (𝑆 ∈ Word 𝑉 → (¬ (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))

Proof of Theorem swrdnd0

Step	Hyp	Ref	Expression
1		ianor 997	. . 3 ⊢ (¬ (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) ↔ (¬ 𝐹 ∈ (0...𝐿) ∨ ¬ 𝐿 ∈ (0...(♯‘𝑆))))
2		3ianor 1124	. . . . 5 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ↔ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿))
3		elfz2nn0 13755	. . . . 5 ⊢ (𝐹 ∈ (0...𝐿) ↔ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿))
4	2, 3	xchnxbir 336	. . . 4 ⊢ (¬ 𝐹 ∈ (0...𝐿) ↔ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿))
5		3ianor 1124	. . . . 5 ⊢ (¬ (𝐿 ∈ ℕ0 ∧ (♯‘𝑆) ∈ ℕ0 ∧ 𝐿 ≤ (♯‘𝑆)) ↔ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆)))
6		elfz2nn0 13755	. . . . 5 ⊢ (𝐿 ∈ (0...(♯‘𝑆)) ↔ (𝐿 ∈ ℕ0 ∧ (♯‘𝑆) ∈ ℕ0 ∧ 𝐿 ≤ (♯‘𝑆)))
7	5, 6	xchnxbir 336	. . . 4 ⊢ (¬ 𝐿 ∈ (0...(♯‘𝑆)) ↔ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆)))
8	4, 7	orbi12i 928	. . 3 ⊢ ((¬ 𝐹 ∈ (0...𝐿) ∨ ¬ 𝐿 ∈ (0...(♯‘𝑆))) ↔ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ∨ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆))))
9	1, 8	bitri 278	. 2 ⊢ (¬ (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) ↔ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ∨ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆))))
10		df-3or 1104	. . . . 5 ⊢ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ↔ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) ∨ ¬ 𝐹 ≤ 𝐿))
11		ianor 997	. . . . . . 7 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ↔ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0))
12		swrdnnn0nd 14810	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝑉 ∧ ¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
13	12	expcom 419	. . . . . . 7 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
14	11, 13	sylbir 238	. . . . . 6 ⊢ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
15		anor 998	. . . . . . . 8 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ↔ ¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0))
16		nn0re 12618	. . . . . . . . . 10 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
17		nn0re 12618	. . . . . . . . . 10 ⊢ (𝐹 ∈ ℕ0 → 𝐹 ∈ ℝ)
18		ltnle 11392	. . . . . . . . . 10 ⊢ ((𝐿 ∈ ℝ ∧ 𝐹 ∈ ℝ) → (𝐿 < 𝐹 ↔ ¬ 𝐹 ≤ 𝐿))
19	16, 17, 18	syl2anr 609	. . . . . . . . 9 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝐿 < 𝐹 ↔ ¬ 𝐹 ≤ 𝐿))
20		nn0z 12720	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ ℕ0 → 𝐹 ∈ ℤ)
21		nn0z 12720	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℤ)
22	20, 21	anim12i 625	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
23	22	anim2i 629	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)))
24		3anass 1111	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ↔ (𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)))
25	23, 24	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
26	25	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) ∧ 𝐿 < 𝐹) → (𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
27	17, 16	anim12ci 626	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝐿 ∈ ℝ ∧ 𝐹 ∈ ℝ))
28	27	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝐿 ∈ ℝ ∧ 𝐹 ∈ ℝ))
29		ltle 11401	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℝ ∧ 𝐹 ∈ ℝ) → (𝐿 < 𝐹 → 𝐿 ≤ 𝐹))
30	28, 29	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝐿 < 𝐹 → 𝐿 ≤ 𝐹))
31	30	imp 412	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) ∧ 𝐿 < 𝐹) → 𝐿 ≤ 𝐹)
32	31	3mix2d 1356	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) ∧ 𝐿 < 𝐹) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿))
33		swrdnd 14808	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
34	26, 32, 33	sylc 66	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) ∧ 𝐿 < 𝐹) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
35	34	ex 418	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝐿 < 𝐹 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
36	35	expcom 419	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑆 ∈ Word 𝑉 → (𝐿 < 𝐹 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
37	36	com23 87	. . . . . . . . 9 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝐿 < 𝐹 → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
38	19, 37	sylbird 263	. . . . . . . 8 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (¬ 𝐹 ≤ 𝐿 → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
39	15, 38	sylbir 238	. . . . . . 7 ⊢ (¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) → (¬ 𝐹 ≤ 𝐿 → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
40	39	imp 412	. . . . . 6 ⊢ ((¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) ∧ ¬ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
41	14, 40	jaoi3 1076	. . . . 5 ⊢ (((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) ∨ ¬ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
42	10, 41	sylbi 220	. . . 4 ⊢ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
43		3anor 1125	. . . . . 6 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ↔ ¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿))
44		pm2.24 125	. . . . . . . . 9 ⊢ (𝐿 ∈ ℕ0 → (¬ 𝐿 ∈ ℕ0 → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
45	44	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → (¬ 𝐿 ∈ ℕ0 → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
46	45	com12 33	. . . . . . 7 ⊢ (¬ 𝐿 ∈ ℕ0 → ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
47		pm2.24 125	. . . . . . . . 9 ⊢ ((♯‘𝑆) ∈ ℕ0 → (¬ (♯‘𝑆) ∈ ℕ0 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
48		lencl 14681	. . . . . . . . 9 ⊢ (𝑆 ∈ Word 𝑉 → (♯‘𝑆) ∈ ℕ0)
49	47, 48	syl11 34	. . . . . . . 8 ⊢ (¬ (♯‘𝑆) ∈ ℕ0 → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
50	49	a1d 26	. . . . . . 7 ⊢ (¬ (♯‘𝑆) ∈ ℕ0 → ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
51	48	nn0red 12671	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝑉 → (♯‘𝑆) ∈ ℝ)
52	16	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → 𝐿 ∈ ℝ)
53		ltnle 11392	. . . . . . . . . . 11 ⊢ (((♯‘𝑆) ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((♯‘𝑆) < 𝐿 ↔ ¬ 𝐿 ≤ (♯‘𝑆)))
54	51, 52, 53	syl2anr 609	. . . . . . . . . 10 ⊢ (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → ((♯‘𝑆) < 𝐿 ↔ ¬ 𝐿 ≤ (♯‘𝑆)))
55		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → 𝑆 ∈ Word 𝑉)
56	20	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → 𝐹 ∈ ℤ)
57	56	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → 𝐹 ∈ ℤ)
58	21	3ad2ant2 1152	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → 𝐿 ∈ ℤ)
59	58	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → 𝐿 ∈ ℤ)
60	55, 57, 59	3jca 1146	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → (𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
61		3mix3 1351	. . . . . . . . . . 11 ⊢ ((♯‘𝑆) < 𝐿 → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿))
62	60, 61, 33	syl2im 41	. . . . . . . . . 10 ⊢ (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → ((♯‘𝑆) < 𝐿 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
63	54, 62	sylbird 263	. . . . . . . . 9 ⊢ (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → (¬ 𝐿 ≤ (♯‘𝑆) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
64	63	com12 33	. . . . . . . 8 ⊢ (¬ 𝐿 ≤ (♯‘𝑆) → (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
65	64	expd 421	. . . . . . 7 ⊢ (¬ 𝐿 ≤ (♯‘𝑆) → ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
66	46, 50, 65	3jaoi 1454	. . . . . 6 ⊢ ((¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆)) → ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
67	43, 66	biimtrrid 246	. . . . 5 ⊢ ((¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆)) → (¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
68	67	impcom 413	. . . 4 ⊢ ((¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ∧ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆))) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
69	42, 68	jaoi3 1076	. . 3 ⊢ (((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ∨ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆))) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
70	69	com12 33	. 2 ⊢ (𝑆 ∈ Word 𝑉 → (((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ∨ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆))) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
71	9, 70	biimtrid 245	1 ⊢ (𝑆 ∈ Word 𝑉 → (¬ (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∅c0 4278  ⟨cop 4834   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  ℝcr 11202  0cc0 11203   < clt 11346   ≤ cle 11347  ℕ₀cn0 12609  ℤcz 12696  ...cfz 13642  ♯chash 14477  Word cword 14661   substr csubstr 14792

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-er 8720  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-card 10023  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-n0 12610  df-z 12697  df-uz 12969  df-fz 13643  df-fzo 13792  df-hash 14478  df-word 14662  df-substr 14793

This theorem is used by:  swrdwrdsymb  14816