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Theorem prub 10214
 Description: A positive fraction not in a positive real is an upper bound. Remark (1) of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
prub (((𝐴P𝐵𝐴) ∧ 𝐶Q) → (¬ 𝐶𝐴𝐵 <Q 𝐶))

Proof of Theorem prub
StepHypRef Expression
1 eleq1 2853 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝐴𝐶𝐴))
21biimpcd 241 . . . . . 6 (𝐵𝐴 → (𝐵 = 𝐶𝐶𝐴))
32adantl 474 . . . . 5 ((𝐴P𝐵𝐴) → (𝐵 = 𝐶𝐶𝐴))
4 prcdnq 10213 . . . . 5 ((𝐴P𝐵𝐴) → (𝐶 <Q 𝐵𝐶𝐴))
53, 4jaod 845 . . . 4 ((𝐴P𝐵𝐴) → ((𝐵 = 𝐶𝐶 <Q 𝐵) → 𝐶𝐴))
65con3d 150 . . 3 ((𝐴P𝐵𝐴) → (¬ 𝐶𝐴 → ¬ (𝐵 = 𝐶𝐶 <Q 𝐵)))
76adantr 473 . 2 (((𝐴P𝐵𝐴) ∧ 𝐶Q) → (¬ 𝐶𝐴 → ¬ (𝐵 = 𝐶𝐶 <Q 𝐵)))
8 elprnq 10211 . . 3 ((𝐴P𝐵𝐴) → 𝐵Q)
9 ltsonq 10189 . . . 4 <Q Or Q
10 sotric 5353 . . . 4 (( <Q Or Q ∧ (𝐵Q𝐶Q)) → (𝐵 <Q 𝐶 ↔ ¬ (𝐵 = 𝐶𝐶 <Q 𝐵)))
119, 10mpan 677 . . 3 ((𝐵Q𝐶Q) → (𝐵 <Q 𝐶 ↔ ¬ (𝐵 = 𝐶𝐶 <Q 𝐵)))
128, 11sylan 572 . 2 (((𝐴P𝐵𝐴) ∧ 𝐶Q) → (𝐵 <Q 𝐶 ↔ ¬ (𝐵 = 𝐶𝐶 <Q 𝐵)))
137, 12sylibrd 251 1 (((𝐴P𝐵𝐴) ∧ 𝐶Q) → (¬ 𝐶𝐴𝐵 <Q 𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∨ wo 833   = wceq 1507   ∈ wcel 2050   class class class wbr 4929   Or wor 5325  Qcnq 10072
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