elicc3 - Mathbox for Jeff Hankins

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Theorem elicc3 36856

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 13420	. 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 13187	. . . . . . 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 36852	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐴 ≤ 𝐵))
8		df-ne 2959	. . . . . . . . . 10 ⊢ (𝐶 ≠ 𝐴 ↔ ¬ 𝐶 = 𝐴)
9		xrleltne 13174	. . . . . . . . . . 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 727	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐴 → 𝐴 < 𝐶))
14		eqcom 2770	. . . . . . . . . . . . . 14 ⊢ (𝐶 = 𝐵 ↔ 𝐵 = 𝐶)
15	14	necon3bbii 3005	. . . . . . . . . . . . 13 ⊢ (¬ 𝐶 = 𝐵 ↔ 𝐵 ≠ 𝐶)
16		xrleltne 13174	. . . . . . . . . . . . . 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 423	. . . . . . . . 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 726	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))
24	13, 23	anim12d 620	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
25	24	ex 417	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))))
26		df-or 861	. . . . . 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 883	. . . . . . . 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 13180	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
38	37	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐴)
39		breq2 5113	. . . . . . . 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 13178	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
42	41	adantr 485	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
43	42	adantllr 731	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
44	43	adantrd 496	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐴 ≤ 𝐶))
45		simpr 489	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
46		breq2 5113	. . . . . . . 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 424	. . . . 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 5112	. . . . . . . 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 13178	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 < 𝐵 → 𝐶 ≤ 𝐵))
54	53	ancoms 463	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 < 𝐵 → 𝐶 ≤ 𝐵))
55	54	adantld 495	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵))
56	55	adantll 726	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵))
57	56	adantr 485	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵))
58		xrleid 13180	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
59	58	ad3antlr 743	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵)
60		breq1 5112	. . . . . . . 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 424	. . . . 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 400 ∨ wo 860 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 class class class wbr 5109 (class class class)co 7410 ℝ^*cxr 11246 < clt 11247 ≤ cle 11248 [,]cicc 13379

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-pre-lttri 11178 ax-pre-lttrn 11179

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-po 5569 df-so 5570 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-icc 13383

This theorem is used by: ivthALT 36874

W3C validator