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Theorem suplocexprlemss 7530
 Description: Lemma for suplocexpr 7540. 𝐴 is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.)
Hypotheses
Ref Expression
suplocexpr.m (𝜑 → ∃𝑥 𝑥𝐴)
suplocexpr.ub (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
suplocexpr.loc (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
Assertion
Ref Expression
suplocexprlemss (𝜑𝐴P)
Distinct variable groups:   𝑥,𝐴,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧)

Proof of Theorem suplocexprlemss
StepHypRef Expression
1 suplocexpr.ub . . 3 (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
2 rsp 2480 . . . . . 6 (∀𝑦𝐴 𝑦<P 𝑥 → (𝑦𝐴𝑦<P 𝑥))
3 ltrelpr 7320 . . . . . . . 8 <P ⊆ (P × P)
43brel 4591 . . . . . . 7 (𝑦<P 𝑥 → (𝑦P𝑥P))
54simpld 111 . . . . . 6 (𝑦<P 𝑥𝑦P)
62, 5syl6 33 . . . . 5 (∀𝑦𝐴 𝑦<P 𝑥 → (𝑦𝐴𝑦P))
76a1i 9 . . . 4 (𝜑 → (∀𝑦𝐴 𝑦<P 𝑥 → (𝑦𝐴𝑦P)))
87rexlimdvw 2553 . . 3 (𝜑 → (∃𝑥P𝑦𝐴 𝑦<P 𝑥 → (𝑦𝐴𝑦P)))
91, 8mpd 13 . 2 (𝜑 → (𝑦𝐴𝑦P))
109ssrdv 3103 1 (𝜑𝐴P)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 697  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417   ⊆ wss 3071   class class class wbr 3929  Pcnp 7106
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