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Theorem xrlttri3 9784
Description: Extended real version of lttri3 8027. (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 9763 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 elxr 9763 . 2 (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3 lttri3 8027 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
43ancoms 268 . . . . 5 ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
5 renepnf 7995 . . . . . . . . . 10 (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
65adantr 276 . . . . . . . . 9 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵 ≠ +∞)
7 neeq2 2361 . . . . . . . . . 10 (𝐴 = +∞ → (𝐵𝐴𝐵 ≠ +∞))
87adantl 277 . . . . . . . . 9 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → (𝐵𝐴𝐵 ≠ +∞))
96, 8mpbird 167 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵𝐴)
109necomd 2433 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐴𝐵)
1110neneqd 2368 . . . . . 6 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → ¬ 𝐴 = 𝐵)
12 ltpnf 9767 . . . . . . . . 9 (𝐵 ∈ ℝ → 𝐵 < +∞)
1312adantr 276 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵 < +∞)
14 breq2 4004 . . . . . . . . 9 (𝐴 = +∞ → (𝐵 < 𝐴𝐵 < +∞))
1514adantl 277 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → (𝐵 < 𝐴𝐵 < +∞))
1613, 15mpbird 167 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
17 notnot 629 . . . . . . . . 9 ((𝐴 < 𝐵𝐵 < 𝐴) → ¬ ¬ (𝐴 < 𝐵𝐵 < 𝐴))
1817olcs 736 . . . . . . . 8 (𝐵 < 𝐴 → ¬ ¬ (𝐴 < 𝐵𝐵 < 𝐴))
19 ioran 752 . . . . . . . 8 (¬ (𝐴 < 𝐵𝐵 < 𝐴) ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
2018, 19sylnib 676 . . . . . . 7 (𝐵 < 𝐴 → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
2116, 20syl 14 . . . . . 6 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
2211, 212falsed 702 . . . . 5 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
23 renemnf 7996 . . . . . . . . . 10 (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
2423adantr 276 . . . . . . . . 9 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐵 ≠ -∞)
25 neeq2 2361 . . . . . . . . . 10 (𝐴 = -∞ → (𝐵𝐴𝐵 ≠ -∞))
2625adantl 277 . . . . . . . . 9 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → (𝐵𝐴𝐵 ≠ -∞))
2724, 26mpbird 167 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐵𝐴)
2827necomd 2433 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐴𝐵)
2928neneqd 2368 . . . . . 6 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → ¬ 𝐴 = 𝐵)
30 mnflt 9770 . . . . . . . . 9 (𝐵 ∈ ℝ → -∞ < 𝐵)
3130adantr 276 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → -∞ < 𝐵)
32 breq1 4003 . . . . . . . . 9 (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
3332adantl 277 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
3431, 33mpbird 167 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐴 < 𝐵)
35 orc 712 . . . . . . 7 (𝐴 < 𝐵 → (𝐴 < 𝐵𝐵 < 𝐴))
36 oranim 781 . . . . . . 7 ((𝐴 < 𝐵𝐵 < 𝐴) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
3734, 35, 363syl 17 . . . . . 6 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
3829, 372falsed 702 . . . . 5 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
394, 22, 383jaodan 1306 . . . 4 ((𝐵 ∈ ℝ ∧ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
4039ancoms 268 . . 3 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
41 renepnf 7995 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
4241adantl 277 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ +∞)
43 neeq2 2361 . . . . . . . . 9 (𝐵 = +∞ → (𝐴𝐵𝐴 ≠ +∞))
4443adantr 276 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → (𝐴𝐵𝐴 ≠ +∞))
4542, 44mpbird 167 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴𝐵)
4645neneqd 2368 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → ¬ 𝐴 = 𝐵)
47 ltpnf 9767 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 < +∞)
4847adantl 277 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴 < +∞)
49 breq2 4004 . . . . . . . . 9 (𝐵 = +∞ → (𝐴 < 𝐵𝐴 < +∞))
5049adantr 276 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → (𝐴 < 𝐵𝐴 < +∞))
5148, 50mpbird 167 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴 < 𝐵)
5251, 35, 363syl 17 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
5346, 522falsed 702 . . . . 5 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
54 eqtr3 2197 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = +∞) → 𝐵 = 𝐴)
5554eqcomd 2183 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = +∞) → 𝐴 = 𝐵)
56 pnfxr 8000 . . . . . . . . 9 +∞ ∈ ℝ*
57 xrltnr 9766 . . . . . . . . 9 (+∞ ∈ ℝ* → ¬ +∞ < +∞)
5856, 57ax-mp 5 . . . . . . . 8 ¬ +∞ < +∞
59 breq12 4005 . . . . . . . . 9 ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ +∞ < +∞))
6059ancoms 268 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 = +∞) → (𝐴 < 𝐵 ↔ +∞ < +∞))
6158, 60mtbiri 675 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = +∞) → ¬ 𝐴 < 𝐵)
62 breq12 4005 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ +∞ < +∞))
6358, 62mtbiri 675 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = +∞) → ¬ 𝐵 < 𝐴)
6461, 63jca 306 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = +∞) → (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
6555, 642thd 175 . . . . 5 ((𝐵 = +∞ ∧ 𝐴 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
66 mnfnepnf 8003 . . . . . . . . 9 -∞ ≠ +∞
67 eqeq12 2190 . . . . . . . . . 10 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 = 𝐵 ↔ -∞ = +∞))
6867necon3bid 2388 . . . . . . . . 9 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴𝐵 ↔ -∞ ≠ +∞))
6966, 68mpbiri 168 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴𝐵)
7069ancoms 268 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = -∞) → 𝐴𝐵)
7170neneqd 2368 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ 𝐴 = 𝐵)
72 mnfltpnf 9772 . . . . . . . . 9 -∞ < +∞
73 breq12 4005 . . . . . . . . 9 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ -∞ < +∞))
7472, 73mpbiri 168 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
7574ancoms 268 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = -∞) → 𝐴 < 𝐵)
7675, 35, 363syl 17 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
7771, 762falsed 702 . . . . 5 ((𝐵 = +∞ ∧ 𝐴 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
7853, 65, 773jaodan 1306 . . . 4 ((𝐵 = +∞ ∧ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
7978ancoms 268 . . 3 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
80 renemnf 7996 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
8180adantl 277 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ -∞)
82 neeq2 2361 . . . . . . . . 9 (𝐵 = -∞ → (𝐴𝐵𝐴 ≠ -∞))
8382adantr 276 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → (𝐴𝐵𝐴 ≠ -∞))
8481, 83mpbird 167 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → 𝐴𝐵)
8584neneqd 2368 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → ¬ 𝐴 = 𝐵)
86 mnflt 9770 . . . . . . . . 9 (𝐴 ∈ ℝ → -∞ < 𝐴)
8786adantl 277 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → -∞ < 𝐴)
88 breq1 4003 . . . . . . . . 9 (𝐵 = -∞ → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
8988adantr 276 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
9087, 89mpbird 167 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → 𝐵 < 𝐴)
9190, 20syl 14 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
9285, 912falsed 702 . . . . 5 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
9366neii 2349 . . . . . . . . . 10 ¬ -∞ = +∞
94 eqeq12 2190 . . . . . . . . . 10 ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 = 𝐴 ↔ -∞ = +∞))
9593, 94mtbiri 675 . . . . . . . . 9 ((𝐵 = -∞ ∧ 𝐴 = +∞) → ¬ 𝐵 = 𝐴)
9695neneqad 2426 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵𝐴)
9796necomd 2433 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐴𝐵)
9897neneqd 2368 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 = +∞) → ¬ 𝐴 = 𝐵)
99 breq12 4005 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ -∞ < +∞))
10072, 99mpbiri 168 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
101100, 20syl 14 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 = +∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
10298, 1012falsed 702 . . . . 5 ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
103 eqtr3 2197 . . . . . . 7 ((𝐴 = -∞ ∧ 𝐵 = -∞) → 𝐴 = 𝐵)
104103ancoms 268 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 = -∞) → 𝐴 = 𝐵)
105 mnfxr 8004 . . . . . . . . 9 -∞ ∈ ℝ*
106 xrltnr 9766 . . . . . . . . 9 (-∞ ∈ ℝ* → ¬ -∞ < -∞)
107105, 106ax-mp 5 . . . . . . . 8 ¬ -∞ < -∞
108 breq12 4005 . . . . . . . . 9 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
109108ancoms 268 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
110107, 109mtbiri 675 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 = -∞) → ¬ 𝐴 < 𝐵)
111 breq12 4005 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 = -∞) → (𝐵 < 𝐴 ↔ -∞ < -∞))
112107, 111mtbiri 675 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 = -∞) → ¬ 𝐵 < 𝐴)
113110, 112jca 306 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 = -∞) → (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
114104, 1132thd 175 . . . . 5 ((𝐵 = -∞ ∧ 𝐴 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
11592, 102, 1143jaodan 1306 . . . 4 ((𝐵 = -∞ ∧ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
116115ancoms 268 . . 3 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
11740, 79, 1163jaodan 1306 . 2 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
1181, 2, 117syl2anb 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 708  w3o 977   = wceq 1353  wcel 2148  wne 2347   class class class wbr 4000  cr 7801  +∞cpnf 7979  -∞cmnf 7980  *cxr 7981   < clt 7982
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-pre-ltirr 7914  ax-pre-apti 7917
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4629  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987
This theorem is referenced by:  xrletri3  9791  iccid  9912  xrmaxleim  11236  xrmaxif  11243  xrmaxaddlem  11252  infxrnegsupex  11255  bdxmet  13668
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