elicc3 - Mathbox for Jeff Hankins

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Theorem elicc3 36751

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 13416	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
2		simp1 1152	. . . . 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 13183	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐴 ≤ 𝐵))
5	4	exp5o 1372	. . . . . 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 36747	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐴 ≤ 𝐵))
8		df-ne 2965	. . . . . . . . . 10 ⊢ (𝐶 ≠ 𝐴 ↔ ¬ 𝐶 = 𝐴)
9		xrleltne 13170	. . . . . . . . . . 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 1201	. . . . . . . 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 2776	. . . . . . . . . . . . . 14 ⊢ (𝐶 = 𝐵 ↔ 𝐵 = 𝐶)
15	14	necon3bbii 3011	. . . . . . . . . . . . 13 ⊢ (¬ 𝐶 = 𝐵 ↔ 𝐵 ≠ 𝐶)
16		xrleltne 13170	. . . . . . . . . . . . . 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 1135	. . . . . . . . . . 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 1187	. . . . . . . 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 1104	. . . . . 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 1145	. . 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 1152	. . . . 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 13176	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
38	37	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐴)
39		breq2 5115	. . . . . . . 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 13174	. . . . . . . . . 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 5115	. . . . . . . 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 1454	. . . . . 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 1365	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → 𝐴 ≤ 𝐶))
51		breq1 5114	. . . . . . . 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 13174	. . . . . . . . . . 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 13176	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
59	58	ad3antlr 743	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵)
60		breq1 5114	. . . . . . . 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 1454	. . . . . 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 1365	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → 𝐶 ≤ 𝐵))
65	36, 50, 64	3jcad 1145	. . 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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∨ w3o 1100 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 class class class wbr 5111 (class class class)co 7411 ℝ^*cxr 11242 < clt 11243 ≤ cle 11244 [,]cicc 13375

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5259 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-pre-lttri 11174 ax-pre-lttrn 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rab 3423 df-v 3463 df-sbc 3752 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5557 df-po 5570 df-so 5571 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 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 7414 df-oprab 7415 df-mpo 7416 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-icc 13379

This theorem is referenced by: ivthALT 36769

W3C validator