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Theorem xrlttri3 9000
Description: Extended real version of lttri3 7310. (Contributed by NM, 9-Feb-2006.)
Assertion
Ref Expression
xrlttri3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))

Proof of Theorem xrlttri3
StepHypRef Expression
1 elxr 8980 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 elxr 8980 . 2 (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3 lttri3 7310 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
43ancoms 264 . . . . 5 ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
5 renepnf 7280 . . . . . . . . . 10 (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
65adantr 270 . . . . . . . . 9 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵 ≠ +∞)
7 neeq2 2263 . . . . . . . . . 10 (𝐴 = +∞ → (𝐵𝐴𝐵 ≠ +∞))
87adantl 271 . . . . . . . . 9 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → (𝐵𝐴𝐵 ≠ +∞))
96, 8mpbird 165 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵𝐴)
109necomd 2335 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐴𝐵)
1110neneqd 2270 . . . . . 6 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → ¬ 𝐴 = 𝐵)
12 ltpnf 8984 . . . . . . . . 9 (𝐵 ∈ ℝ → 𝐵 < +∞)
1312adantr 270 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵 < +∞)
14 breq2 3809 . . . . . . . . 9 (𝐴 = +∞ → (𝐵 < 𝐴𝐵 < +∞))
1514adantl 271 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → (𝐵 < 𝐴𝐵 < +∞))
1613, 15mpbird 165 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
17 notnot 592 . . . . . . . . 9 ((𝐴 < 𝐵𝐵 < 𝐴) → ¬ ¬ (𝐴 < 𝐵𝐵 < 𝐴))
1817olcs 688 . . . . . . . 8 (𝐵 < 𝐴 → ¬ ¬ (𝐴 < 𝐵𝐵 < 𝐴))
19 ioran 702 . . . . . . . 8 (¬ (𝐴 < 𝐵𝐵 < 𝐴) ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
2018, 19sylnib 634 . . . . . . 7 (𝐵 < 𝐴 → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
2116, 20syl 14 . . . . . 6 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
2211, 212falsed 651 . . . . 5 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
23 renemnf 7281 . . . . . . . . . 10 (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
2423adantr 270 . . . . . . . . 9 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐵 ≠ -∞)
25 neeq2 2263 . . . . . . . . . 10 (𝐴 = -∞ → (𝐵𝐴𝐵 ≠ -∞))
2625adantl 271 . . . . . . . . 9 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → (𝐵𝐴𝐵 ≠ -∞))
2724, 26mpbird 165 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐵𝐴)
2827necomd 2335 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐴𝐵)
2928neneqd 2270 . . . . . 6 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → ¬ 𝐴 = 𝐵)
30 mnflt 8986 . . . . . . . . 9 (𝐵 ∈ ℝ → -∞ < 𝐵)
3130adantr 270 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → -∞ < 𝐵)
32 breq1 3808 . . . . . . . . 9 (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
3332adantl 271 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
3431, 33mpbird 165 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐴 < 𝐵)
35 orc 666 . . . . . . 7 (𝐴 < 𝐵 → (𝐴 < 𝐵𝐵 < 𝐴))
36 oranim 841 . . . . . . 7 ((𝐴 < 𝐵𝐵 < 𝐴) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
3734, 35, 363syl 17 . . . . . 6 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
3829, 372falsed 651 . . . . 5 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
394, 22, 383jaodan 1238 . . . 4 ((𝐵 ∈ ℝ ∧ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
4039ancoms 264 . . 3 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
41 renepnf 7280 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
4241adantl 271 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ +∞)
43 neeq2 2263 . . . . . . . . 9 (𝐵 = +∞ → (𝐴𝐵𝐴 ≠ +∞))
4443adantr 270 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → (𝐴𝐵𝐴 ≠ +∞))
4542, 44mpbird 165 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴𝐵)
4645neneqd 2270 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → ¬ 𝐴 = 𝐵)
47 ltpnf 8984 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 < +∞)
4847adantl 271 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴 < +∞)
49 breq2 3809 . . . . . . . . 9 (𝐵 = +∞ → (𝐴 < 𝐵𝐴 < +∞))
5049adantr 270 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → (𝐴 < 𝐵𝐴 < +∞))
5148, 50mpbird 165 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴 < 𝐵)
5251, 35, 363syl 17 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
5346, 522falsed 651 . . . . 5 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
54 eqtr3 2102 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = +∞) → 𝐵 = 𝐴)
5554eqcomd 2088 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = +∞) → 𝐴 = 𝐵)
56 pnfxr 7285 . . . . . . . . 9 +∞ ∈ ℝ*
57 xrltnr 8983 . . . . . . . . 9 (+∞ ∈ ℝ* → ¬ +∞ < +∞)
5856, 57ax-mp 7 . . . . . . . 8 ¬ +∞ < +∞
59 breq12 3810 . . . . . . . . 9 ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ +∞ < +∞))
6059ancoms 264 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 = +∞) → (𝐴 < 𝐵 ↔ +∞ < +∞))
6158, 60mtbiri 633 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = +∞) → ¬ 𝐴 < 𝐵)
62 breq12 3810 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ +∞ < +∞))
6358, 62mtbiri 633 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = +∞) → ¬ 𝐵 < 𝐴)
6461, 63jca 300 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = +∞) → (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
6555, 642thd 173 . . . . 5 ((𝐵 = +∞ ∧ 𝐴 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
66 mnfnepnf 7288 . . . . . . . . 9 -∞ ≠ +∞
67 eqeq12 2095 . . . . . . . . . 10 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 = 𝐵 ↔ -∞ = +∞))
6867necon3bid 2290 . . . . . . . . 9 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴𝐵 ↔ -∞ ≠ +∞))
6966, 68mpbiri 166 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴𝐵)
7069ancoms 264 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = -∞) → 𝐴𝐵)
7170neneqd 2270 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ 𝐴 = 𝐵)
72 mnfltpnf 8988 . . . . . . . . 9 -∞ < +∞
73 breq12 3810 . . . . . . . . 9 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ -∞ < +∞))
7472, 73mpbiri 166 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
7574ancoms 264 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = -∞) → 𝐴 < 𝐵)
7675, 35, 363syl 17 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
7771, 762falsed 651 . . . . 5 ((𝐵 = +∞ ∧ 𝐴 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
7853, 65, 773jaodan 1238 . . . 4 ((𝐵 = +∞ ∧ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
7978ancoms 264 . . 3 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
80 renemnf 7281 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
8180adantl 271 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ -∞)
82 neeq2 2263 . . . . . . . . 9 (𝐵 = -∞ → (𝐴𝐵𝐴 ≠ -∞))
8382adantr 270 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → (𝐴𝐵𝐴 ≠ -∞))
8481, 83mpbird 165 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → 𝐴𝐵)
8584neneqd 2270 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → ¬ 𝐴 = 𝐵)
86 mnflt 8986 . . . . . . . . 9 (𝐴 ∈ ℝ → -∞ < 𝐴)
8786adantl 271 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → -∞ < 𝐴)
88 breq1 3808 . . . . . . . . 9 (𝐵 = -∞ → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
8988adantr 270 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
9087, 89mpbird 165 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → 𝐵 < 𝐴)
9190, 20syl 14 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
9285, 912falsed 651 . . . . 5 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
9366neii 2251 . . . . . . . . . 10 ¬ -∞ = +∞
94 eqeq12 2095 . . . . . . . . . 10 ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 = 𝐴 ↔ -∞ = +∞))
9593, 94mtbiri 633 . . . . . . . . 9 ((𝐵 = -∞ ∧ 𝐴 = +∞) → ¬ 𝐵 = 𝐴)
9695neneqad 2328 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵𝐴)
9796necomd 2335 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐴𝐵)
9897neneqd 2270 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 = +∞) → ¬ 𝐴 = 𝐵)
99 breq12 3810 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ -∞ < +∞))
10072, 99mpbiri 166 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
101100, 20syl 14 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 = +∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
10298, 1012falsed 651 . . . . 5 ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
103 eqtr3 2102 . . . . . . 7 ((𝐴 = -∞ ∧ 𝐵 = -∞) → 𝐴 = 𝐵)
104103ancoms 264 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 = -∞) → 𝐴 = 𝐵)
105 mnfxr 7289 . . . . . . . . 9 -∞ ∈ ℝ*
106 xrltnr 8983 . . . . . . . . 9 (-∞ ∈ ℝ* → ¬ -∞ < -∞)
107105, 106ax-mp 7 . . . . . . . 8 ¬ -∞ < -∞
108 breq12 3810 . . . . . . . . 9 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
109108ancoms 264 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
110107, 109mtbiri 633 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 = -∞) → ¬ 𝐴 < 𝐵)
111 breq12 3810 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 = -∞) → (𝐵 < 𝐴 ↔ -∞ < -∞))
112107, 111mtbiri 633 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 = -∞) → ¬ 𝐵 < 𝐴)
113110, 112jca 300 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 = -∞) → (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
114104, 1132thd 173 . . . . 5 ((𝐵 = -∞ ∧ 𝐴 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
11592, 102, 1143jaodan 1238 . . . 4 ((𝐵 = -∞ ∧ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
116115ancoms 264 . . 3 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
11740, 79, 1163jaodan 1238 . 2 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
1181, 2, 117syl2anb 285 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  w3o 919   = wceq 1285  wcel 1434  wne 2249   class class class wbr 3805  cr 7094  +∞cpnf 7264  -∞cmnf 7265  *cxr 7266   < clt 7267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-pre-ltirr 7202  ax-pre-apti 7205
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-xp 4397  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272
This theorem is referenced by:  xrletri3  9003  iccid  9076
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