elicc3 - Mathbox for Jeff Hankins

	Mathbox for Jeff Hankins		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > elicc3		Structured version Visualization version GIF version

Theorem elicc3 34433

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 13052	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
2		simp1 1134	. . . . 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 12821	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐴 ≤ 𝐵))
5	4	exp5o 1353	. . . . . 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 34428	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐴 ≤ 𝐵))
8		df-ne 2943	. . . . . . . . . 10 ⊢ (𝐶 ≠ 𝐴 ↔ ¬ 𝐶 = 𝐴)
9		xrleltne 12808	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶) → (𝐴 < 𝐶 ↔ 𝐶 ≠ 𝐴))
10	9	biimprd 247	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶) → (𝐶 ≠ 𝐴 → 𝐴 < 𝐶))
11	8, 10	syl5bir 242	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶) → (¬ 𝐶 = 𝐴 → 𝐴 < 𝐶))
12	11	3adant3r3 1182	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐴 → 𝐴 < 𝐶))
13	12	adantlr 711	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐴 → 𝐴 < 𝐶))
14		eqcom 2745	. . . . . . . . . . . . . 14 ⊢ (𝐶 = 𝐵 ↔ 𝐵 = 𝐶)
15	14	necon3bbii 2990	. . . . . . . . . . . . 13 ⊢ (¬ 𝐶 = 𝐵 ↔ 𝐵 ≠ 𝐶)
16		xrleltne 12808	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵) → (𝐶 < 𝐵 ↔ 𝐵 ≠ 𝐶))
17	16	biimprd 247	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵) → (𝐵 ≠ 𝐶 → 𝐶 < 𝐵))
18	15, 17	syl5bi 241	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))
19	18	3exp 1117	. . . . . . . . . . 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 418	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ^* ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))
22	21	3adantr2 1168	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))
23	22	adantll 710	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵))
24	13, 23	anim12d 608	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
25	24	ex 412	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))))
26		df-or 844	. . . . . 6 ⊢ ((𝐶 = 𝐴 ∨ ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) ↔ (¬ 𝐶 = 𝐴 → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))
27		3orass 1088	. . . . . 6 ⊢ ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) ↔ (𝐶 = 𝐴 ∨ ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))
28		pm5.6 998	. . . . . . 7 ⊢ (((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ (¬ 𝐶 = 𝐴 → (𝐶 = 𝐵 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))))
29		orcom 866	. . . . . . . 8 ⊢ ((𝐶 = 𝐵 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵))
30	29	imbi2i 335	. . . . . . 7 ⊢ ((¬ 𝐶 = 𝐴 → (𝐶 = 𝐵 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) ↔ (¬ 𝐶 = 𝐴 → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))
31	28, 30	bitri 274	. . . . . 6 ⊢ (((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ (¬ 𝐶 = 𝐴 → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))
32	26, 27, 31	3bitr4ri 303	. . . . 5 ⊢ (((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵))
33	25, 32	syl6ib 250	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))
34	3, 7, 33	3jcad 1127	. . 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 1134	. . . . 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 12814	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
38	37	ad3antrrr 726	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐴)
39		breq2 5074	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → (𝐴 ≤ 𝐶 ↔ 𝐴 ≤ 𝐴))
40	38, 39	syl5ibrcom 246	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐴 → 𝐴 ≤ 𝐶))
41		xrltle 12812	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
42	41	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
43	42	adantllr 715	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶))
44	43	adantrd 491	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐴 ≤ 𝐶))
45		simpr 484	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
46		breq2 5074	. . . . . . . 8 ⊢ (𝐶 = 𝐵 → (𝐴 ≤ 𝐶 ↔ 𝐴 ≤ 𝐵))
47	45, 46	syl5ibrcom 246	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐵 → 𝐴 ≤ 𝐶))
48	40, 44, 47	3jaod 1426	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐴 ≤ 𝐶))
49	48	exp31 419	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ ℝ^ → (𝐴 ≤ 𝐵 → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐴 ≤ 𝐶))))
50	49	3impd 1346	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → 𝐴 ≤ 𝐶))
51		breq1 5073	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → (𝐶 ≤ 𝐵 ↔ 𝐴 ≤ 𝐵))
52	45, 51	syl5ibrcom 246	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐴 → 𝐶 ≤ 𝐵))
53		xrltle 12812	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 < 𝐵 → 𝐶 ≤ 𝐵))
54	53	ancoms 458	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 < 𝐵 → 𝐶 ≤ 𝐵))
55	54	adantld 490	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵))
56	55	adantll 710	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵))
57	56	adantr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵))
58		xrleid 12814	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
59	58	ad3antlr 727	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵)
60		breq1 5073	. . . . . . . 8 ⊢ (𝐶 = 𝐵 → (𝐶 ≤ 𝐵 ↔ 𝐵 ≤ 𝐵))
61	59, 60	syl5ibrcom 246	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐵 → 𝐶 ≤ 𝐵))
62	52, 57, 61	3jaod 1426	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 ≤ 𝐵) → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐶 ≤ 𝐵))
63	62	exp31 419	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ ℝ^ → (𝐴 ≤ 𝐵 → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐶 ≤ 𝐵))))
64	63	3impd 1346	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → 𝐶 ≤ 𝐵))
65	36, 50, 64	3jcad 1127	. . 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 211	. 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 278	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 205 ∧ wa 395 ∨ wo 843 ∨ w3o 1084 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 class class class wbr 5070 (class class class)co 7255 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 [,]cicc 13011

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-pre-lttri 10876 ax-pre-lttrn 10877

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-icc 13015

This theorem is referenced by: ivthALT 34451

W3C validator