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Theorem xrlttri3 8818
Description: Extended real version of lttri3 7156. (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 8796 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 elxr 8796 . 2 (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3 lttri3 7156 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
43ancoms 259 . . . . 5 ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
5 renepnf 7131 . . . . . . . . . 10 (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
65adantr 265 . . . . . . . . 9 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵 ≠ +∞)
7 neeq2 2234 . . . . . . . . . 10 (𝐴 = +∞ → (𝐵𝐴𝐵 ≠ +∞))
87adantl 266 . . . . . . . . 9 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → (𝐵𝐴𝐵 ≠ +∞))
96, 8mpbird 160 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵𝐴)
109necomd 2306 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐴𝐵)
1110neneqd 2241 . . . . . 6 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → ¬ 𝐴 = 𝐵)
12 ltpnf 8802 . . . . . . . . 9 (𝐵 ∈ ℝ → 𝐵 < +∞)
1312adantr 265 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵 < +∞)
14 breq2 3795 . . . . . . . . 9 (𝐴 = +∞ → (𝐵 < 𝐴𝐵 < +∞))
1514adantl 266 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → (𝐵 < 𝐴𝐵 < +∞))
1613, 15mpbird 160 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
17 notnot 569 . . . . . . . . 9 ((𝐴 < 𝐵𝐵 < 𝐴) → ¬ ¬ (𝐴 < 𝐵𝐵 < 𝐴))
1817olcs 665 . . . . . . . 8 (𝐵 < 𝐴 → ¬ ¬ (𝐴 < 𝐵𝐵 < 𝐴))
19 ioran 679 . . . . . . . 8 (¬ (𝐴 < 𝐵𝐵 < 𝐴) ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
2018, 19sylnib 611 . . . . . . 7 (𝐵 < 𝐴 → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
2116, 20syl 14 . . . . . 6 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
2211, 212falsed 628 . . . . 5 ((𝐵 ∈ ℝ ∧ 𝐴 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
23 renemnf 7132 . . . . . . . . . 10 (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
2423adantr 265 . . . . . . . . 9 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐵 ≠ -∞)
25 neeq2 2234 . . . . . . . . . 10 (𝐴 = -∞ → (𝐵𝐴𝐵 ≠ -∞))
2625adantl 266 . . . . . . . . 9 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → (𝐵𝐴𝐵 ≠ -∞))
2724, 26mpbird 160 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐵𝐴)
2827necomd 2306 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐴𝐵)
2928neneqd 2241 . . . . . 6 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → ¬ 𝐴 = 𝐵)
30 mnflt 8804 . . . . . . . . 9 (𝐵 ∈ ℝ → -∞ < 𝐵)
3130adantr 265 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → -∞ < 𝐵)
32 breq1 3794 . . . . . . . . 9 (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
3332adantl 266 . . . . . . . 8 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
3431, 33mpbird 160 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → 𝐴 < 𝐵)
35 orc 643 . . . . . . 7 (𝐴 < 𝐵 → (𝐴 < 𝐵𝐵 < 𝐴))
36 oranim 818 . . . . . . 7 ((𝐴 < 𝐵𝐵 < 𝐴) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
3734, 35, 363syl 17 . . . . . 6 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
3829, 372falsed 628 . . . . 5 ((𝐵 ∈ ℝ ∧ 𝐴 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
394, 22, 383jaodan 1212 . . . 4 ((𝐵 ∈ ℝ ∧ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
4039ancoms 259 . . 3 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
41 renepnf 7131 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
4241adantl 266 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ +∞)
43 neeq2 2234 . . . . . . . . 9 (𝐵 = +∞ → (𝐴𝐵𝐴 ≠ +∞))
4443adantr 265 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → (𝐴𝐵𝐴 ≠ +∞))
4542, 44mpbird 160 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴𝐵)
4645neneqd 2241 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → ¬ 𝐴 = 𝐵)
47 ltpnf 8802 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 < +∞)
4847adantl 266 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴 < +∞)
49 breq2 3795 . . . . . . . . 9 (𝐵 = +∞ → (𝐴 < 𝐵𝐴 < +∞))
5049adantr 265 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → (𝐴 < 𝐵𝐴 < +∞))
5148, 50mpbird 160 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → 𝐴 < 𝐵)
5251, 35, 363syl 17 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
5346, 522falsed 628 . . . . 5 ((𝐵 = +∞ ∧ 𝐴 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
54 eqtr3 2075 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = +∞) → 𝐵 = 𝐴)
5554eqcomd 2061 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = +∞) → 𝐴 = 𝐵)
56 pnfxr 8792 . . . . . . . . 9 +∞ ∈ ℝ*
57 xrltnr 8801 . . . . . . . . 9 (+∞ ∈ ℝ* → ¬ +∞ < +∞)
5856, 57ax-mp 7 . . . . . . . 8 ¬ +∞ < +∞
59 breq12 3796 . . . . . . . . 9 ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ +∞ < +∞))
6059ancoms 259 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 = +∞) → (𝐴 < 𝐵 ↔ +∞ < +∞))
6158, 60mtbiri 610 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = +∞) → ¬ 𝐴 < 𝐵)
62 breq12 3796 . . . . . . . 8 ((𝐵 = +∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ +∞ < +∞))
6358, 62mtbiri 610 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = +∞) → ¬ 𝐵 < 𝐴)
6461, 63jca 294 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = +∞) → (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
6555, 642thd 168 . . . . 5 ((𝐵 = +∞ ∧ 𝐴 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
66 mnfnepnf 8798 . . . . . . . . 9 -∞ ≠ +∞
67 eqeq12 2068 . . . . . . . . . 10 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 = 𝐵 ↔ -∞ = +∞))
6867necon3bid 2261 . . . . . . . . 9 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴𝐵 ↔ -∞ ≠ +∞))
6966, 68mpbiri 161 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴𝐵)
7069ancoms 259 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = -∞) → 𝐴𝐵)
7170neneqd 2241 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ 𝐴 = 𝐵)
72 mnfltpnf 8806 . . . . . . . . 9 -∞ < +∞
73 breq12 3796 . . . . . . . . 9 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ -∞ < +∞))
7472, 73mpbiri 161 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
7574ancoms 259 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = -∞) → 𝐴 < 𝐵)
7675, 35, 363syl 17 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
7771, 762falsed 628 . . . . 5 ((𝐵 = +∞ ∧ 𝐴 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
7853, 65, 773jaodan 1212 . . . 4 ((𝐵 = +∞ ∧ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
7978ancoms 259 . . 3 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
80 renemnf 7132 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
8180adantl 266 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ -∞)
82 neeq2 2234 . . . . . . . . 9 (𝐵 = -∞ → (𝐴𝐵𝐴 ≠ -∞))
8382adantr 265 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → (𝐴𝐵𝐴 ≠ -∞))
8481, 83mpbird 160 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → 𝐴𝐵)
8584neneqd 2241 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → ¬ 𝐴 = 𝐵)
86 mnflt 8804 . . . . . . . . 9 (𝐴 ∈ ℝ → -∞ < 𝐴)
8786adantl 266 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → -∞ < 𝐴)
88 breq1 3794 . . . . . . . . 9 (𝐵 = -∞ → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
8988adantr 265 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
9087, 89mpbird 160 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → 𝐵 < 𝐴)
9190, 20syl 14 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
9285, 912falsed 628 . . . . 5 ((𝐵 = -∞ ∧ 𝐴 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
9366neii 2222 . . . . . . . . . 10 ¬ -∞ = +∞
94 eqeq12 2068 . . . . . . . . . 10 ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 = 𝐴 ↔ -∞ = +∞))
9593, 94mtbiri 610 . . . . . . . . 9 ((𝐵 = -∞ ∧ 𝐴 = +∞) → ¬ 𝐵 = 𝐴)
9695neneqad 2299 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵𝐴)
9796necomd 2306 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐴𝐵)
9897neneqd 2241 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 = +∞) → ¬ 𝐴 = 𝐵)
99 breq12 3796 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ -∞ < +∞))
10072, 99mpbiri 161 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
101100, 20syl 14 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 = +∞) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
10298, 1012falsed 628 . . . . 5 ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
103 eqtr3 2075 . . . . . . 7 ((𝐴 = -∞ ∧ 𝐵 = -∞) → 𝐴 = 𝐵)
104103ancoms 259 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 = -∞) → 𝐴 = 𝐵)
105 mnfxr 8794 . . . . . . . . 9 -∞ ∈ ℝ*
106 xrltnr 8801 . . . . . . . . 9 (-∞ ∈ ℝ* → ¬ -∞ < -∞)
107105, 106ax-mp 7 . . . . . . . 8 ¬ -∞ < -∞
108 breq12 3796 . . . . . . . . 9 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
109108ancoms 259 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
110107, 109mtbiri 610 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 = -∞) → ¬ 𝐴 < 𝐵)
111 breq12 3796 . . . . . . . 8 ((𝐵 = -∞ ∧ 𝐴 = -∞) → (𝐵 < 𝐴 ↔ -∞ < -∞))
112107, 111mtbiri 610 . . . . . . 7 ((𝐵 = -∞ ∧ 𝐴 = -∞) → ¬ 𝐵 < 𝐴)
113110, 112jca 294 . . . . . 6 ((𝐵 = -∞ ∧ 𝐴 = -∞) → (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
114104, 1132thd 168 . . . . 5 ((𝐵 = -∞ ∧ 𝐴 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
11592, 102, 1143jaodan 1212 . . . 4 ((𝐵 = -∞ ∧ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
116115ancoms 259 . . 3 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 = -∞) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
11740, 79, 1163jaodan 1212 . 2 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
1181, 2, 117syl2anb 279 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639  w3o 895   = wceq 1259  wcel 1409  wne 2220   class class class wbr 3791  cr 6945  +∞cpnf 7115  -∞cmnf 7116  *cxr 7117   < clt 7118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-cnex 7032  ax-resscn 7033  ax-pre-ltirr 7053  ax-pre-apti 7056
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-xp 4378  df-pnf 7120  df-mnf 7121  df-xr 7122  df-ltxr 7123
This theorem is referenced by:  xrletri3  8821  iccid  8894
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