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Theorem xrlttri3 9863

Description: Extended real version of lttri3 8099. (Contributed by NM, 9-Feb-2006.)

**Assertion**
Ref	Expression
xrlttri3	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))

**Proof of Theorem xrlttri3**
Step	Hyp	Ref	Expression
1		elxr 9842	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 9842	. 2 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		lttri3 8099	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
4	3	ancoms 268	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
5		renepnf 8067	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
6	5	adantr 276	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵 ≠ +∞)
7		neeq2 2378	. . . . . . . . . 10 ⊢ (𝐴 = +∞ → (𝐵 ≠ 𝐴 ↔ 𝐵 ≠ +∞))
8	7	adantl 277	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → (𝐵 ≠ 𝐴 ↔ 𝐵 ≠ +∞))
9	6, 8	mpbird 167	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵 ≠ 𝐴)
10	9	necomd 2450	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐴 ≠ 𝐵)
11	10	neneqd 2385	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → ¬ 𝐴 = 𝐵)
12		ltpnf 9846	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
13	12	adantr 276	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵 < +∞)
14		breq2 4033	. . . . . . . . 9 ⊢ (𝐴 = +∞ → (𝐵 < 𝐴 ↔ 𝐵 < +∞))
15	14	adantl 277	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ 𝐵 < +∞))
16	13, 15	mpbird 167	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
17		notnot 630	. . . . . . . . 9 ⊢ ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) → ¬ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
18	17	olcs 737	. . . . . . . 8 ⊢ (𝐵 < 𝐴 → ¬ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
19		ioran 753	. . . . . . . 8 ⊢ (¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
20	18, 19	sylnib 677	. . . . . . 7 ⊢ (𝐵 < 𝐴 → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
21	16, 20	syl 14	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
22	11, 21	2falsed 703	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
23		renemnf 8068	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
24	23	adantr 276	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐵 ≠ -∞)
25		neeq2 2378	. . . . . . . . . 10 ⊢ (𝐴 = -∞ → (𝐵 ≠ 𝐴 ↔ 𝐵 ≠ -∞))
26	25	adantl 277	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → (𝐵 ≠ 𝐴 ↔ 𝐵 ≠ -∞))
27	24, 26	mpbird 167	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐵 ≠ 𝐴)
28	27	necomd 2450	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐴 ≠ 𝐵)
29	28	neneqd 2385	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → ¬ 𝐴 = 𝐵)
30		mnflt 9849	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵)
31	30	adantr 276	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → -∞ < 𝐵)
32		breq1 4032	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
33	32	adantl 277	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
34	31, 33	mpbird 167	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐴 < 𝐵)
35		orc 713	. . . . . . 7 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
36		oranim 782	. . . . . . 7 ⊢ ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
37	34, 35, 36	3syl 17	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
38	29, 37	2falsed 703	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
39	4, 22, 38	3jaodan 1317	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
40	39	ancoms 268	. . 3 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
41		renepnf 8067	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
42	41	adantl 277	. . . . . . . 8 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ +∞)
43		neeq2 2378	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ +∞))
44	43	adantr 276	. . . . . . . 8 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ +∞))
45	42, 44	mpbird 167	. . . . . . 7 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ 𝐵)
46	45	neneqd 2385	. . . . . 6 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → ¬ 𝐴 = 𝐵)
47		ltpnf 9846	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
48	47	adantl 277	. . . . . . . 8 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴 < +∞)
49		breq2 4033	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (𝐴 < 𝐵 ↔ 𝐴 < +∞))
50	49	adantr 276	. . . . . . . 8 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 < +∞))
51	48, 50	mpbird 167	. . . . . . 7 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴 < 𝐵)
52	51, 35, 36	3syl 17	. . . . . 6 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
53	46, 52	2falsed 703	. . . . 5 ⊢ ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
54		eqtr3 2213	. . . . . . 7 ⊢ ((𝐵 = +∞ ∧ 𝐴 = +∞) → 𝐵 = 𝐴)
55	54	eqcomd 2199	. . . . . 6 ⊢ ((𝐵 = +∞ ∧ 𝐴 = +∞) → 𝐴 = 𝐵)
56		pnfxr 8072	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
57		xrltnr 9845	. . . . . . . . 9 ⊢ (+∞ ∈ ℝ^* → ¬ +∞ < +∞)
58	56, 57	ax-mp 5	. . . . . . . 8 ⊢ ¬ +∞ < +∞
59		breq12 4034	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ +∞ < +∞))
60	59	ancoms 268	. . . . . . . 8 ⊢ ((𝐵 = +∞ ∧ 𝐴 = +∞) → (𝐴 < 𝐵 ↔ +∞ < +∞))
61	58, 60	mtbiri 676	. . . . . . 7 ⊢ ((𝐵 = +∞ ∧ 𝐴 = +∞) → ¬ 𝐴 < 𝐵)
62		breq12 4034	. . . . . . . 8 ⊢ ((𝐵 = +∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ +∞ < +∞))
63	58, 62	mtbiri 676	. . . . . . 7 ⊢ ((𝐵 = +∞ ∧ 𝐴 = +∞) → ¬ 𝐵 < 𝐴)
64	61, 63	jca 306	. . . . . 6 ⊢ ((𝐵 = +∞ ∧ 𝐴 = +∞) → (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
65	55, 64	2thd 175	. . . . 5 ⊢ ((𝐵 = +∞ ∧ 𝐴 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
66		mnfnepnf 8075	. . . . . . . . 9 ⊢ -∞ ≠ +∞
67		eqeq12 2206	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 = 𝐵 ↔ -∞ = +∞))
68	67	necon3bid 2405	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 ≠ 𝐵 ↔ -∞ ≠ +∞))
69	66, 68	mpbiri 168	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴 ≠ 𝐵)
70	69	ancoms 268	. . . . . . 7 ⊢ ((𝐵 = +∞ ∧ 𝐴 = -∞) → 𝐴 ≠ 𝐵)
71	70	neneqd 2385	. . . . . 6 ⊢ ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ 𝐴 = 𝐵)
72		mnfltpnf 9851	. . . . . . . . 9 ⊢ -∞ < +∞
73		breq12 4034	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ -∞ < +∞))
74	72, 73	mpbiri 168	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
75	74	ancoms 268	. . . . . . 7 ⊢ ((𝐵 = +∞ ∧ 𝐴 = -∞) → 𝐴 < 𝐵)
76	75, 35, 36	3syl 17	. . . . . 6 ⊢ ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
77	71, 76	2falsed 703	. . . . 5 ⊢ ((𝐵 = +∞ ∧ 𝐴 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
78	53, 65, 77	3jaodan 1317	. . . 4 ⊢ ((𝐵 = +∞ ∧ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
79	78	ancoms 268	. . 3 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
80		renemnf 8068	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
81	80	adantl 277	. . . . . . . 8 ⊢ ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ -∞)
82		neeq2 2378	. . . . . . . . 9 ⊢ (𝐵 = -∞ → (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ -∞))
83	82	adantr 276	. . . . . . . 8 ⊢ ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ -∞))
84	81, 83	mpbird 167	. . . . . . 7 ⊢ ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ 𝐵)
85	84	neneqd 2385	. . . . . 6 ⊢ ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → ¬ 𝐴 = 𝐵)
86		mnflt 9849	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
87	86	adantl 277	. . . . . . . 8 ⊢ ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → -∞ < 𝐴)
88		breq1 4032	. . . . . . . . 9 ⊢ (𝐵 = -∞ → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
89	88	adantr 276	. . . . . . . 8 ⊢ ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
90	87, 89	mpbird 167	. . . . . . 7 ⊢ ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → 𝐵 < 𝐴)
91	90, 20	syl 14	. . . . . 6 ⊢ ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
92	85, 91	2falsed 703	. . . . 5 ⊢ ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
93	66	neii 2366	. . . . . . . . . 10 ⊢ ¬ -∞ = +∞
94		eqeq12 2206	. . . . . . . . . 10 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 = 𝐴 ↔ -∞ = +∞))
95	93, 94	mtbiri 676	. . . . . . . . 9 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → ¬ 𝐵 = 𝐴)
96	95	neneqad 2443	. . . . . . . 8 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵 ≠ 𝐴)
97	96	necomd 2450	. . . . . . 7 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐴 ≠ 𝐵)
98	97	neneqd 2385	. . . . . 6 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → ¬ 𝐴 = 𝐵)
99		breq12 4034	. . . . . . . 8 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ -∞ < +∞))
100	72, 99	mpbiri 168	. . . . . . 7 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
101	100, 20	syl 14	. . . . . 6 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
102	98, 101	2falsed 703	. . . . 5 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
103		eqtr3 2213	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → 𝐴 = 𝐵)
104	103	ancoms 268	. . . . . 6 ⊢ ((𝐵 = -∞ ∧ 𝐴 = -∞) → 𝐴 = 𝐵)
105		mnfxr 8076	. . . . . . . . 9 ⊢ -∞ ∈ ℝ^*
106		xrltnr 9845	. . . . . . . . 9 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
107	105, 106	ax-mp 5	. . . . . . . 8 ⊢ ¬ -∞ < -∞
108		breq12 4034	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
109	108	ancoms 268	. . . . . . . 8 ⊢ ((𝐵 = -∞ ∧ 𝐴 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
110	107, 109	mtbiri 676	. . . . . . 7 ⊢ ((𝐵 = -∞ ∧ 𝐴 = -∞) → ¬ 𝐴 < 𝐵)
111		breq12 4034	. . . . . . . 8 ⊢ ((𝐵 = -∞ ∧ 𝐴 = -∞) → (𝐵 < 𝐴 ↔ -∞ < -∞))
112	107, 111	mtbiri 676	. . . . . . 7 ⊢ ((𝐵 = -∞ ∧ 𝐴 = -∞) → ¬ 𝐵 < 𝐴)
113	110, 112	jca 306	. . . . . 6 ⊢ ((𝐵 = -∞ ∧ 𝐴 = -∞) → (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
114	104, 113	2thd 175	. . . . 5 ⊢ ((𝐵 = -∞ ∧ 𝐴 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
115	92, 102, 114	3jaodan 1317	. . . 4 ⊢ ((𝐵 = -∞ ∧ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
116	115	ancoms 268	. . 3 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
117	40, 79, 116	3jaodan 1317	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
118	1, 2, 117	syl2anb 291	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 ∨ w3o 979 = wceq 1364 ∈ wcel 2164 ≠ wne 2364 class class class wbr 4029 ℝcr 7871 +∞cpnf 8051 -∞cmnf 8052 ℝ^*cxr 8053 < clt 8054

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-pre-ltirr 7984 ax-pre-apti 7987

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-xp 4665 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059

This theorem is referenced by: xrletri3 9870 iccid 9991 xrmaxleim 11387 xrmaxif 11394 xrmaxaddlem 11403 infxrnegsupex 11406 bdxmet 14669

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