Theorem swrdnd0 14719

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 13665	. . . . 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 13665	. . . . 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 14718	. . . . . . . 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 12531	. . . . . . . . . 10 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
17		nn0re 12531	. . . . . . . . . 10 ⊢ (𝐹 ∈ ℕ0 → 𝐹 ∈ ℝ)
18		ltnle 11307	. . . . . . . . . 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 12633	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ ℕ0 → 𝐹 ∈ ℤ)
21		nn0z 12633	. . . . . . . . . . . . . . . . 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 11316	. . . . . . . . . . . . . . . 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 14716	. . . . . . . . . . . . 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 14590	. . . . . . . . 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 12584	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝑉 → (♯‘𝑆) ∈ ℝ)
52	16	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → 𝐿 ∈ ℝ)
53		ltnle 11307	. . . . . . . . . . 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 1570   ∈ wcel 2146  ∅c0 4289  ⟨cop 4600   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423  ℝcr 11117  0cc0 11118   < clt 11261   ≤ cle 11262  ℕ₀cn0 12522  ℤcz 12609  ...cfz 13553  ♯chash 14386  Word cword 14570   substr csubstr 14700

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-card 9944  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-n0 12523  df-z 12610  df-uz 12881  df-fz 13554  df-fzo 13702  df-hash 14387  df-word 14571  df-substr 14701

This theorem is used by:  swrdwrdsymb  14724