Theorem swrdnd0 14680

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 983	. . 3 ⊢ (¬ (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) ↔ (¬ 𝐹 ∈ (0...𝐿) ∨ ¬ 𝐿 ∈ (0...(♯‘𝑆))))
2		3ianor 1106	. . . . 5 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) ↔ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿))
3		elfz2nn0 13640	. . . . 5 ⊢ (𝐹 ∈ (0...𝐿) ↔ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿))
4	2, 3	xchnxbir 333	. . . 4 ⊢ (¬ 𝐹 ∈ (0...𝐿) ↔ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿))
5		3ianor 1106	. . . . 5 ⊢ (¬ (𝐿 ∈ ℕ0 ∧ (♯‘𝑆) ∈ ℕ0 ∧ 𝐿 ≤ (♯‘𝑆)) ↔ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆)))
6		elfz2nn0 13640	. . . . 5 ⊢ (𝐿 ∈ (0...(♯‘𝑆)) ↔ (𝐿 ∈ ℕ0 ∧ (♯‘𝑆) ∈ ℕ0 ∧ 𝐿 ≤ (♯‘𝑆)))
7	5, 6	xchnxbir 333	. . . 4 ⊢ (¬ 𝐿 ∈ (0...(♯‘𝑆)) ↔ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆)))
8	4, 7	orbi12i 914	. . 3 ⊢ ((¬ 𝐹 ∈ (0...𝐿) ∨ ¬ 𝐿 ∈ (0...(♯‘𝑆))) ↔ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ∨ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆))))
9	1, 8	bitri 275	. 2 ⊢ (¬ (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) ↔ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ∨ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆))))
10		df-3or 1087	. . . . 5 ⊢ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ↔ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) ∨ ¬ 𝐹 ≤ 𝐿))
11		ianor 983	. . . . . . 7 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ↔ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0))
12		swrdnnn0nd 14679	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝑉 ∧ ¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
13	12	expcom 413	. . . . . . 7 ⊢ (¬ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
14	11, 13	sylbir 235	. . . . . 6 ⊢ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
15		anor 984	. . . . . . . 8 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) ↔ ¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0))
16		nn0re 12515	. . . . . . . . . 10 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
17		nn0re 12515	. . . . . . . . . 10 ⊢ (𝐹 ∈ ℕ0 → 𝐹 ∈ ℝ)
18		ltnle 11319	. . . . . . . . . 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 12618	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ ℕ0 → 𝐹 ∈ ℤ)
21		nn0z 12618	. . . . . . . . . . . . . . . . 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 1094	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ↔ (𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ)))
25	23, 24	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝑆 ∈ Word 𝑉 ∧ 𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ))
26	25	adantr 480	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝐿 ∈ ℝ ∧ 𝐹 ∈ ℝ))
29		ltle 11328	. . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) ∧ 𝐿 < 𝐹) → 𝐿 ≤ 𝐹)
32	31	3mix2d 1338	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) ∧ 𝐿 < 𝐹) → (𝐹 < 0 ∨ 𝐿 ≤ 𝐹 ∨ (♯‘𝑆) < 𝐿))
33		swrdnd 14677	. . . . . . . . . . . . 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 412	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝐿 < 𝐹 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
36	35	expcom 413	. . . . . . . . . 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 260	. . . . . . . 8 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (¬ 𝐹 ≤ 𝐿 → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
39	15, 38	sylbir 235	. . . . . . 7 ⊢ (¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) → (¬ 𝐹 ≤ 𝐿 → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
40	39	imp 406	. . . . . 6 ⊢ ((¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) ∧ ¬ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
41	14, 40	jaoi3 1060	. . . . 5 ⊢ (((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0) ∨ ¬ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
42	10, 41	sylbi 217	. . . 4 ⊢ ((¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
43		3anor 1107	. . . . . 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 14556	. . . . . . . . 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 12568	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝑉 → (♯‘𝑆) ∈ ℝ)
52	16	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → 𝐿 ∈ ℝ)
53		ltnle 11319	. . . . . . . . . . 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 484	. . . . . . . . . . . 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 480	. . . . . . . . . . . 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 480	. . . . . . . . . . . 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 1333	. . . . . . . . . . 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 260	. . . . . . . . 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 415	. . . . . . 7 ⊢ (¬ 𝐿 ≤ (♯‘𝑆) → ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
66	46, 50, 65	3jaoi 1430	. . . . . 6 ⊢ ((¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆)) → ((𝐹 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
67	43, 66	biimtrrid 243	. . . . 5 ⊢ ((¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆)) → (¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)))
68	67	impcom 407	. . . 4 ⊢ ((¬ (¬ 𝐹 ∈ ℕ0 ∨ ¬ 𝐿 ∈ ℕ0 ∨ ¬ 𝐹 ≤ 𝐿) ∧ (¬ 𝐿 ∈ ℕ0 ∨ ¬ (♯‘𝑆) ∈ ℕ0 ∨ ¬ 𝐿 ≤ (♯‘𝑆))) → (𝑆 ∈ Word 𝑉 → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
69	42, 68	jaoi3 1060	. . 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 242	1 ⊢ (𝑆 ∈ Word 𝑉 → (¬ (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∅c0 4313  ⟨cop 4612   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  ℝcr 11133  0cc0 11134   < clt 11274   ≤ cle 11275  ℕ₀cn0 12506  ℤcz 12593  ...cfz 13529  ♯chash 14353  Word cword 14536   substr csubstr 14663

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-n0 12507  df-z 12594  df-uz 12858  df-fz 13530  df-fzo 13677  df-hash 14354  df-word 14537  df-substr 14664

This theorem is referenced by:  swrdwrdsymb  14685