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Theorem ltsopr 9892
 Description: Positive real 'less than' is a strict ordering. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltsopr <P Or P

Proof of Theorem ltsopr
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pssirr 3740 . . . 4 ¬ 𝑥𝑥
2 ltprord 9890 . . . 4 ((𝑥P𝑥P) → (𝑥<P 𝑥𝑥𝑥))
31, 2mtbiri 316 . . 3 ((𝑥P𝑥P) → ¬ 𝑥<P 𝑥)
43anidms 678 . 2 (𝑥P → ¬ 𝑥<P 𝑥)
5 psstr 3744 . . 3 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
6 ltprord 9890 . . . . . 6 ((𝑥P𝑦P) → (𝑥<P 𝑦𝑥𝑦))
763adant3 1101 . . . . 5 ((𝑥P𝑦P𝑧P) → (𝑥<P 𝑦𝑥𝑦))
8 ltprord 9890 . . . . . 6 ((𝑦P𝑧P) → (𝑦<P 𝑧𝑦𝑧))
983adant1 1099 . . . . 5 ((𝑥P𝑦P𝑧P) → (𝑦<P 𝑧𝑦𝑧))
107, 9anbi12d 747 . . . 4 ((𝑥P𝑦P𝑧P) → ((𝑥<P 𝑦𝑦<P 𝑧) ↔ (𝑥𝑦𝑦𝑧)))
11 ltprord 9890 . . . . 5 ((𝑥P𝑧P) → (𝑥<P 𝑧𝑥𝑧))
12113adant2 1100 . . . 4 ((𝑥P𝑦P𝑧P) → (𝑥<P 𝑧𝑥𝑧))
1310, 12imbi12d 333 . . 3 ((𝑥P𝑦P𝑧P) → (((𝑥<P 𝑦𝑦<P 𝑧) → 𝑥<P 𝑧) ↔ ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)))
145, 13mpbiri 248 . 2 ((𝑥P𝑦P𝑧P) → ((𝑥<P 𝑦𝑦<P 𝑧) → 𝑥<P 𝑧))
15 psslinpr 9891 . . 3 ((𝑥P𝑦P) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
16 biidd 252 . . . 4 ((𝑥P𝑦P) → (𝑥 = 𝑦𝑥 = 𝑦))
17 ltprord 9890 . . . . 5 ((𝑦P𝑥P) → (𝑦<P 𝑥𝑦𝑥))
1817ancoms 468 . . . 4 ((𝑥P𝑦P) → (𝑦<P 𝑥𝑦𝑥))
196, 16, 183orbi123d 1438 . . 3 ((𝑥P𝑦P) → ((𝑥<P 𝑦𝑥 = 𝑦𝑦<P 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
2015, 19mpbird 247 . 2 ((𝑥P𝑦P) → (𝑥<P 𝑦𝑥 = 𝑦𝑦<P 𝑥))
214, 14, 20issoi 5095 1 <P Or P
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∨ w3o 1053   ∧ w3a 1054   ∈ wcel 2030   ⊊ wpss 3608   class class class wbr 4685   Or wor 5063  Pcnp 9719
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