Theorem swrdnd0 14603

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 980	. . 3 ⊢ (¬ (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) ↔ (¬ 𝐹 ∈ (0...𝐿) ∨ ¬ 𝐿 ∈ (0...(♯‘𝑆))))
2		3ianor 1107	. . . . 5 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ↔ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿))
3		elfz2nn0 13588	. . . . 5 ⊢ (𝐹 ∈ (0...𝐿) ↔ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿))
4	2, 3	xchnxbir 332	. . . 4 ⊢ (¬ 𝐹 ∈ (0...𝐿) ↔ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿))
5		3ianor 1107	. . . . 5 ⊢ (¬ (𝐿 ∈ ℕ0 ∧ (♯‘𝑆) ∈ ℕ0 ∧ 𝐿 ≤ (♯‘𝑆)) ↔ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆)))
6		elfz2nn0 13588	. . . . 5 ⊢ (𝐿 ∈ (0...(♯‘𝑆)) ↔ (𝐿 ∈ ℕ0 ∧ (♯‘𝑆) ∈ ℕ0 ∧ 𝐿 ≤ (♯‘𝑆)))
7	5, 6	xchnxbir 332	. . . 4 ⊢ (¬ 𝐿 ∈ (0...(♯‘𝑆)) ↔ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆)))
8	4, 7	orbi12i 913	. . 3 ⊢ ((¬ 𝐹 ∈ (0...𝐿) ∨ ¬ 𝐿 ∈ (0...(♯‘𝑆))) ↔ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ∨ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆))))
9	1, 8	bitri 274	. 2 ⊢ (¬ (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) ↔ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ∨ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆))))
10		df-3or 1088	. . . . 5 ⊢ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ↔ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) ∨ ¬ 𝐹 ≤ 𝐿))
11		ianor 980	. . . . . . 7 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ↔ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0))
12		swrdnnn0nd 14602	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝑉 ∧ ¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
13	12	expcom 414	. . . . . . 7 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
14	11, 13	sylbir 234	. . . . . 6 ⊢ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
15		anor 981	. . . . . . . 8 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ↔ ¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0))
16		nn0re 12477	. . . . . . . . . 10 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
17		nn0re 12477	. . . . . . . . . 10 ⊢ (𝐹 ∈ ℕ0 → 𝐹 ∈ ℝ)
18		ltnle 11289	. . . . . . . . . 10 ⊢ ((𝐿 ∈ ℝ ∧ 𝐹 ∈ ℝ) → (𝐿 < 𝐹 ↔ ¬ 𝐹 ≤ 𝐿))
19	16, 17, 18	syl2anr 597	. . . . . . . . 9 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝐿 < 𝐹 ↔ ¬ 𝐹 ≤ 𝐿))
20		nn0z 12579	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ ℕ0 → 𝐹 ∈ ℤ)
21		nn0z 12579	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℤ)
22	20, 21	anim12i 613	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
23	22	anim2i 617	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)))
24		3anass 1095	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ↔ (𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)))
25	23, 24	sylibr 233	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
26	25	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) ∧ 𝐿 < 𝐹) → (𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
27	17, 16	anim12ci 614	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝐿 ∈ ℝ ∧ 𝐹 ∈ ℝ))
28	27	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝐿 ∈ ℝ ∧ 𝐹 ∈ ℝ))
29		ltle 11298	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℝ ∧ 𝐹 ∈ ℝ) → (𝐿 < 𝐹 → 𝐿 ≤ 𝐹))
30	28, 29	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝐿 < 𝐹 → 𝐿 ≤ 𝐹))
31	30	imp 407	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) ∧ 𝐿 < 𝐹) → 𝐿 ≤ 𝐹)
32	31	3mix2d 1337	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) ∧ 𝐿 < 𝐹) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿))
33		swrdnd 14600	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
34	26, 32, 33	sylc 65	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) ∧ 𝐿 < 𝐹) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
35	34	ex 413	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝐿 < 𝐹 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
36	35	expcom 414	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑆 ∈ Word 𝑉 → (𝐿 < 𝐹 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
37	36	com23 86	. . . . . . . . 9 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝐿 < 𝐹 → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
38	19, 37	sylbird 259	. . . . . . . 8 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (¬ 𝐹 ≤ 𝐿 → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
39	15, 38	sylbir 234	. . . . . . 7 ⊢ (¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) → (¬ 𝐹 ≤ 𝐿 → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
40	39	imp 407	. . . . . 6 ⊢ ((¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) ∧ ¬ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
41	14, 40	jaoi3 1059	. . . . 5 ⊢ (((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) ∨ ¬ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
42	10, 41	sylbi 216	. . . 4 ⊢ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
43		3anor 1108	. . . . . 6 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ↔ ¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿))
44		pm2.24 124	. . . . . . . . 9 ⊢ (𝐿 ∈ ℕ0 → (¬ 𝐿 ∈ ℕ0 → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
45	44	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → (¬ 𝐿 ∈ ℕ0 → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
46	45	com12 32	. . . . . . 7 ⊢ (¬ 𝐿 ∈ ℕ0 → ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
47		pm2.24 124	. . . . . . . . 9 ⊢ ((♯‘𝑆) ∈ ℕ0 → (¬ (♯‘𝑆) ∈ ℕ0 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
48		lencl 14479	. . . . . . . . 9 ⊢ (𝑆 ∈ Word 𝑉 → (♯‘𝑆) ∈ ℕ0)
49	47, 48	syl11 33	. . . . . . . 8 ⊢ (¬ (♯‘𝑆) ∈ ℕ0 → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
50	49	a1d 25	. . . . . . 7 ⊢ (¬ (♯‘𝑆) ∈ ℕ0 → ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
51	48	nn0red 12529	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝑉 → (♯‘𝑆) ∈ ℝ)
52	16	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → 𝐿 ∈ ℝ)
53		ltnle 11289	. . . . . . . . . . 11 ⊢ (((♯‘𝑆) ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((♯‘𝑆) < 𝐿 ↔ ¬ 𝐿 ≤ (♯‘𝑆)))
54	51, 52, 53	syl2anr 597	. . . . . . . . . 10 ⊢ (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → ((♯‘𝑆) < 𝐿 ↔ ¬ 𝐿 ≤ (♯‘𝑆)))
55		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → 𝑆 ∈ Word 𝑉)
56	20	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → 𝐹 ∈ ℤ)
57	56	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → 𝐹 ∈ ℤ)
58	21	3ad2ant2 1134	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → 𝐿 ∈ ℤ)
59	58	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → 𝐿 ∈ ℤ)
60	55, 57, 59	3jca 1128	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → (𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
61		3mix3 1332	. . . . . . . . . . 11 ⊢ ((♯‘𝑆) < 𝐿 → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿))
62	60, 61, 33	syl2im 40	. . . . . . . . . 10 ⊢ (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → ((♯‘𝑆) < 𝐿 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
63	54, 62	sylbird 259	. . . . . . . . 9 ⊢ (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → (¬ 𝐿 ≤ (♯‘𝑆) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
64	63	com12 32	. . . . . . . 8 ⊢ (¬ 𝐿 ≤ (♯‘𝑆) → (((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ∧ 𝑆 ∈ Word 𝑉) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
65	64	expd 416	. . . . . . 7 ⊢ (¬ 𝐿 ≤ (♯‘𝑆) → ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
66	46, 50, 65	3jaoi 1427	. . . . . 6 ⊢ ((¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆)) → ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
67	43, 66	biimtrrid 242	. . . . 5 ⊢ ((¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆)) → (¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
68	67	impcom 408	. . . 4 ⊢ ((¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ∧ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆))) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
69	42, 68	jaoi3 1059	. . 3 ⊢ (((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ∨ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆))) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
70	69	com12 32	. 2 ⊢ (𝑆 ∈ Word 𝑉 → (((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ∨ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆))) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
71	9, 70	biimtrid 241	1 ⊢ (𝑆 ∈ Word 𝑉 → (¬ (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∨ w3o 1086   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∅c0 4321  ⟨cop 4633   class class class wbr 5147  ‘cfv 6540  (class class class)co 7405  ℝcr 11105  0cc0 11106   < clt 11244   ≤ cle 11245  ℕ₀cn0 12468  ℤcz 12554  ...cfz 13480  ♯chash 14286  Word cword 14460   substr csubstr 14586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-substr 14587

This theorem is referenced by:  swrdwrdsymb  14608