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Mirrors > Home > MPE Home > Th. List > prnmax | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
prnmax | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 <Q 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2817 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
2 | 1 | anbi2d 629 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ P ∧ 𝑦 ∈ 𝐴) ↔ (𝐴 ∈ P ∧ 𝐵 ∈ 𝐴))) |
3 | breq1 5151 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 <Q 𝑥 ↔ 𝐵 <Q 𝑥)) | |
4 | 3 | rexbidv 3175 | . . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 <Q 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝐵 <Q 𝑥)) |
5 | 2, 4 | imbi12d 344 | . . 3 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ P ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝑦 <Q 𝑥) ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 <Q 𝑥))) |
6 | elnpi 11012 | . . . . . 6 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑦 ∈ 𝐴 (∀𝑥(𝑥 <Q 𝑦 → 𝑥 ∈ 𝐴) ∧ ∃𝑥 ∈ 𝐴 𝑦 <Q 𝑥))) | |
7 | 6 | simprbi 496 | . . . . 5 ⊢ (𝐴 ∈ P → ∀𝑦 ∈ 𝐴 (∀𝑥(𝑥 <Q 𝑦 → 𝑥 ∈ 𝐴) ∧ ∃𝑥 ∈ 𝐴 𝑦 <Q 𝑥)) |
8 | 7 | r19.21bi 3245 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ 𝐴) → (∀𝑥(𝑥 <Q 𝑦 → 𝑥 ∈ 𝐴) ∧ ∃𝑥 ∈ 𝐴 𝑦 <Q 𝑥)) |
9 | 8 | simprd 495 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝑦 <Q 𝑥) |
10 | 5, 9 | vtoclg 3540 | . 2 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 <Q 𝑥)) |
11 | 10 | anabsi7 670 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 <Q 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 ∀wal 1532 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ∃wrex 3067 Vcvv 3471 ⊊ wpss 3948 ∅c0 4323 class class class wbr 5148 Qcnq 10876 <Q cltq 10882 Pcnp 10883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-np 11005 |
This theorem is referenced by: npomex 11020 prnmadd 11021 genpnmax 11031 1idpr 11053 ltexprlem4 11063 reclem3pr 11073 suplem1pr 11076 |
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