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Theorem prnmax 9696
 Description: A positive real has no largest member. Definition 9-3.1(iii) of [Gleason] p. 121. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
prnmax ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem prnmax
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2676 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐴𝐵𝐴))
21anbi2d 736 . . . 4 (𝑦 = 𝐵 → ((𝐴P𝑦𝐴) ↔ (𝐴P𝐵𝐴)))
3 breq1 4586 . . . . 5 (𝑦 = 𝐵 → (𝑦 <Q 𝑥𝐵 <Q 𝑥))
43rexbidv 3034 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 <Q 𝑥 ↔ ∃𝑥𝐴 𝐵 <Q 𝑥))
52, 4imbi12d 333 . . 3 (𝑦 = 𝐵 → (((𝐴P𝑦𝐴) → ∃𝑥𝐴 𝑦 <Q 𝑥) ↔ ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥)))
6 elnpi 9689 . . . . . 6 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑦𝐴 (∀𝑥(𝑥 <Q 𝑦𝑥𝐴) ∧ ∃𝑥𝐴 𝑦 <Q 𝑥)))
76simprbi 479 . . . . 5 (𝐴P → ∀𝑦𝐴 (∀𝑥(𝑥 <Q 𝑦𝑥𝐴) ∧ ∃𝑥𝐴 𝑦 <Q 𝑥))
87r19.21bi 2916 . . . 4 ((𝐴P𝑦𝐴) → (∀𝑥(𝑥 <Q 𝑦𝑥𝐴) ∧ ∃𝑥𝐴 𝑦 <Q 𝑥))
98simprd 478 . . 3 ((𝐴P𝑦𝐴) → ∃𝑥𝐴 𝑦 <Q 𝑥)
105, 9vtoclg 3239 . 2 (𝐵𝐴 → ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥))
1110anabsi7 856 1 ((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊊ wpss 3541  ∅c0 3874   class class class wbr 4583  Qcnq 9553
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