elicc3 - Mathbox for Jeff Hankins

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Theorem elicc3 36560

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 13337	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
2		simp1 1143	. . . . 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 13104	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐴 ≤ 𝐵))
5	4	exp5o 1363	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐶 ∈ ℝ^* → (𝐵 ∈ ℝ^* → (𝐴 ≤ 𝐶 → (𝐶 ≤ 𝐵 → 𝐴 ≤ 𝐵)))))
6	5	com23 86	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (𝐵 ∈ ℝ^* → (𝐶 ∈ ℝ^* → (𝐴 ≤ 𝐶 → (𝐶 ≤ 𝐵 → 𝐴 ≤ 𝐵)))))
7	6	imp5q 36555	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐴 ≤ 𝐵))
8		df-ne 2937	. . . . . . . . . 10 ⊢ (𝐶 ≠ 𝐴 ↔ ¬ 𝐶 = 𝐴)
9		xrleltne 13091	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶) → (𝐴 < 𝐶 ↔ 𝐶 ≠ 𝐴))
10	9	biimprd 250	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶) → (𝐶 ≠ 𝐴 → 𝐴 < 𝐶))
11	8, 10	biimtrrid 245	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶) → (¬ 𝐶 = 𝐴 → 𝐴 < 𝐶))
12	11	3adant3r3 1192	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐴 → 𝐴 < 𝐶))
13	12	adantlr 722	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐴 → 𝐴 < 𝐶))
14		eqcom 2748	. . . . . . . . . . . . . 14 ⊢ (𝐶 = 𝐵 ↔ 𝐵 = 𝐶)
15	14	necon3bbii 2983	. . . . . . . . . . . . 13 ⊢ (¬ 𝐶 = 𝐵 ↔ 𝐵 ≠ 𝐶)
16		xrleltne 13091	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵) → (𝐶 < 𝐵 ↔ 𝐵 ≠ 𝐶))
17	16	biimprd 250	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵) → (𝐵 ≠ 𝐶 → 𝐶 < 𝐵))
18	15, 17	biimtrid 244	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))
19	18	3exp 1126	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℝ^* → (𝐵 ∈ ℝ^* → (𝐶 ≤ 𝐵 → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))))
20	19	com12 32	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → (𝐶 ∈ ℝ^* → (𝐶 ≤ 𝐵 → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))))
21	20	imp32 420	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))
22	21	3adantr2 1178	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))
23	22	adantll 721	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))
24	13, 23	anim12d 616	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
25	24	ex 414	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))))
26		df-or 855	. . . . . 6 ⊢ ((𝐶 = 𝐴 ∨ ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) ↔ (¬ 𝐶 = 𝐴 → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))
27		3orass 1096	. . . . . 6 ⊢ ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) ↔ (𝐶 = 𝐴 ∨ ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))
28		pm5.6 1010	. . . . . . 7 ⊢ (((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ (¬ 𝐶 = 𝐴 → (𝐶 = 𝐵 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))))
29		orcom 877	. . . . . . . 8 ⊢ ((𝐶 = 𝐵 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵))
30	29	imbi2i 338	. . . . . . 7 ⊢ ((¬ 𝐶 = 𝐴 → (𝐶 = 𝐵 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) ↔ (¬ 𝐶 = 𝐴 → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))
31	28, 30	bitri 277	. . . . . 6 ⊢ (((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ (¬ 𝐶 = 𝐴 → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))
32	26, 27, 31	3bitr4ri 306	. . . . 5 ⊢ (((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵))
33	25, 32	imbitrdi 253	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))
34	3, 7, 33	3jcad 1136	. . 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 1143	. . . . 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 13097	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
38	37	ad3antrrr 737	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐴)
39		breq2 5079	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → (𝐴 ≤ 𝐶 ↔ 𝐴 ≤ 𝐴))
40	38, 39	syl5ibrcom 249	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐴 → 𝐴 ≤ 𝐶))
41		xrltle 13095	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
42	41	adantr 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
43	42	adantllr 726	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
44	43	adantrd 493	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐴 ≤ 𝐶))
45		simpr 486	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
46		breq2 5079	. . . . . . . 8 ⊢ (𝐶 = 𝐵 → (𝐴 ≤ 𝐶 ↔ 𝐴 ≤ 𝐵))
47	45, 46	syl5ibrcom 249	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐵 → 𝐴 ≤ 𝐶))
48	40, 44, 47	3jaod 1438	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐴 ≤ 𝐶))
49	48	exp31 421	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ ℝ^ → (𝐴 ≤ 𝐵 → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐴 ≤ 𝐶))))
50	49	3impd 1356	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → 𝐴 ≤ 𝐶))
51		breq1 5078	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → (𝐶 ≤ 𝐵 ↔ 𝐴 ≤ 𝐵))
52	45, 51	syl5ibrcom 249	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐴 → 𝐶 ≤ 𝐵))
53		xrltle 13095	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 < 𝐵 → 𝐶 ≤ 𝐵))
54	53	ancoms 460	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 < 𝐵 → 𝐶 ≤ 𝐵))
55	54	adantld 492	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵))
56	55	adantll 721	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵))
57	56	adantr 482	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵))
58		xrleid 13097	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
59	58	ad3antlr 738	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵)
60		breq1 5078	. . . . . . . 8 ⊢ (𝐶 = 𝐵 → (𝐶 ≤ 𝐵 ↔ 𝐵 ≤ 𝐵))
61	59, 60	syl5ibrcom 249	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐵 → 𝐶 ≤ 𝐵))
62	52, 57, 61	3jaod 1438	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐶 ≤ 𝐵))
63	62	exp31 421	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ ℝ^ → (𝐴 ≤ 𝐵 → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐶 ≤ 𝐵))))
64	63	3impd 1356	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → 𝐶 ≤ 𝐵))
65	36, 50, 64	3jcad 1136	. . 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 214	. 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 281	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 208 ∧ wa 397 ∨ wo 854 ∨ w3o 1092 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 class class class wbr 5075 (class class class)co 7360 ℝ^*cxr 11173 < clt 11174 ≤ cle 11175 [,]cicc 13296

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-pre-lttri 11107 ax-pre-lttrn 11108

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-br 5076 df-opab 5138 df-mpt 5157 df-id 5516 df-po 5529 df-so 5530 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-ov 7363 df-oprab 7364 df-mpo 7365 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-icc 13300

This theorem is referenced by: ivthALT 36578

W3C validator