elicc3 - Mathbox for Jeff Hankins

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Theorem elicc3 36921

Description: An equivalent membership condition for closed intervals. (Contributed by Jeff Hankins, 14-Jul-2009.)

**Assertion**
Ref	Expression
elicc3	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵))))

**Proof of Theorem elicc3**
Step	Hyp	Ref	Expression
1		elicc1 13442	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
2		simp1 1154	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐶 ∈ ℝ^*)
3	2	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐶 ∈ ℝ^*))
4		xrletr 13209	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐴 ≤ 𝐵))
5	4	exp5o 1374	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐶 ∈ ℝ^* → (𝐵 ∈ ℝ^* → (𝐴 ≤ 𝐶 → (𝐶 ≤ 𝐵 → 𝐴 ≤ 𝐵)))))
6	5	com23 87	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (𝐵 ∈ ℝ^* → (𝐶 ∈ ℝ^* → (𝐴 ≤ 𝐶 → (𝐶 ≤ 𝐵 → 𝐴 ≤ 𝐵)))))
7	6	imp5q 36917	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐴 ≤ 𝐵))
8		df-ne 2958	. . . . . . . . . 10 ⊢ (𝐶 ≠ 𝐴 ↔ ¬ 𝐶 = 𝐴)
9		xrleltne 13196	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶) → (𝐴 < 𝐶 ↔ 𝐶 ≠ 𝐴))
10	9	biimprd 251	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶) → (𝐶 ≠ 𝐴 → 𝐴 < 𝐶))
11	8, 10	biimtrrid 246	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶) → (¬ 𝐶 = 𝐴 → 𝐴 < 𝐶))
12	11	3adant3r3 1203	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐴 → 𝐴 < 𝐶))
13	12	adantlr 728	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐴 → 𝐴 < 𝐶))
14		eqcom 2769	. . . . . . . . . . . . . 14 ⊢ (𝐶 = 𝐵 ↔ 𝐵 = 𝐶)
15	14	necon3bbii 3004	. . . . . . . . . . . . 13 ⊢ (¬ 𝐶 = 𝐵 ↔ 𝐵 ≠ 𝐶)
16		xrleltne 13196	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵) → (𝐶 < 𝐵 ↔ 𝐵 ≠ 𝐶))
17	16	biimprd 251	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵) → (𝐵 ≠ 𝐶 → 𝐶 < 𝐵))
18	15, 17	biimtrid 245	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))
19	18	3exp 1137	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℝ^* → (𝐵 ∈ ℝ^* → (𝐶 ≤ 𝐵 → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))))
20	19	com12 33	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → (𝐶 ∈ ℝ^* → (𝐶 ≤ 𝐵 → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))))
21	20	imp32 424	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))
22	21	3adantr2 1189	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))
23	22	adantll 727	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))
24	13, 23	anim12d 621	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
25	24	ex 418	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))))
26		df-or 862	. . . . . 6 ⊢ ((𝐶 = 𝐴 ∨ ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) ↔ (¬ 𝐶 = 𝐴 → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))
27		3orass 1106	. . . . . 6 ⊢ ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) ↔ (𝐶 = 𝐴 ∨ ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))
28		pm5.6 1017	. . . . . . 7 ⊢ (((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ (¬ 𝐶 = 𝐴 → (𝐶 = 𝐵 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))))
29		orcom 884	. . . . . . . 8 ⊢ ((𝐶 = 𝐵 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵))
30	29	imbi2i 339	. . . . . . 7 ⊢ ((¬ 𝐶 = 𝐴 → (𝐶 = 𝐵 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) ↔ (¬ 𝐶 = 𝐴 → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))
31	28, 30	bitri 278	. . . . . 6 ⊢ (((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ (¬ 𝐶 = 𝐴 → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))
32	26, 27, 31	3bitr4ri 307	. . . . 5 ⊢ (((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵))
33	25, 32	imbitrdi 254	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))
34	3, 7, 33	3jcad 1147	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵))))
35		simp1 1154	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → 𝐶 ∈ ℝ^*)
36	35	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → 𝐶 ∈ ℝ^*))
37		xrleid 13202	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
38	37	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐴)
39		breq2 5111	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → (𝐴 ≤ 𝐶 ↔ 𝐴 ≤ 𝐴))
40	38, 39	syl5ibrcom 250	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐴 → 𝐴 ≤ 𝐶))
41		xrltle 13200	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
42	41	adantr 486	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
43	42	adantllr 732	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
44	43	adantrd 497	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐴 ≤ 𝐶))
45		simpr 490	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
46		breq2 5111	. . . . . . . 8 ⊢ (𝐶 = 𝐵 → (𝐴 ≤ 𝐶 ↔ 𝐴 ≤ 𝐵))
47	45, 46	syl5ibrcom 250	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐵 → 𝐴 ≤ 𝐶))
48	40, 44, 47	3jaod 1456	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐴 ≤ 𝐶))
49	48	exp31 425	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ ℝ^ → (𝐴 ≤ 𝐵 → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐴 ≤ 𝐶))))
50	49	3impd 1367	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → 𝐴 ≤ 𝐶))
51		breq1 5110	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → (𝐶 ≤ 𝐵 ↔ 𝐴 ≤ 𝐵))
52	45, 51	syl5ibrcom 250	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐴 → 𝐶 ≤ 𝐵))
53		xrltle 13200	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 < 𝐵 → 𝐶 ≤ 𝐵))
54	53	ancoms 464	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 < 𝐵 → 𝐶 ≤ 𝐵))
55	54	adantld 496	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵))
56	55	adantll 727	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵))
57	56	adantr 486	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵))
58		xrleid 13202	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
59	58	ad3antlr 744	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵)
60		breq1 5110	. . . . . . . 8 ⊢ (𝐶 = 𝐵 → (𝐶 ≤ 𝐵 ↔ 𝐵 ≤ 𝐵))
61	59, 60	syl5ibrcom 250	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐵 → 𝐶 ≤ 𝐵))
62	52, 57, 61	3jaod 1456	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐶 ≤ 𝐵))
63	62	exp31 425	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ ℝ^ → (𝐴 ≤ 𝐵 → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐶 ≤ 𝐵))))
64	63	3impd 1367	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → 𝐶 ≤ 𝐵))
65	36, 50, 64	3jcad 1147	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
66	34, 65	impbid 215	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵))))
67	1, 66	bitrd 282	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 class class class wbr 5107 (class class class)co 7416 ℝ^*cxr 11267 < clt 11268 ≤ cle 11269 [,]cicc 13401

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11181 ax-resscn 11182 ax-pre-lttri 11199 ax-pre-lttrn 11200

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-po 5567 df-so 5568 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-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7419 df-oprab 7420 df-mpo 7421 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-icc 13405

This theorem is used by: ivthALT 36939

W3C validator