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Theorem prnmax 10916

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	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 <_Q 𝑥)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

Proof of Theorem prnmax Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2828	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
2	1	anbi2d 636	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ P ∧ 𝑦 ∈ 𝐴) ↔ (𝐴 ∈ P ∧ 𝐵 ∈ 𝐴)))
3		breq1 5082	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 <_Q 𝑥 ↔ 𝐵 <_Q 𝑥))
4	3	rexbidv 3164	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 <_Q 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝐵 <_Q 𝑥))
5	2, 4	imbi12d 345	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ P ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝑦 <_Q 𝑥) ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 <_Q 𝑥)))
6		elnpi 10909	. . . . . 6 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑦 ∈ 𝐴 (∀𝑥(𝑥 <_Q 𝑦 → 𝑥 ∈ 𝐴) ∧ ∃𝑥 ∈ 𝐴 𝑦 <_Q 𝑥)))
7	6	simprbi 498	. . . . 5 ⊢ (𝐴 ∈ P → ∀𝑦 ∈ 𝐴 (∀𝑥(𝑥 <_Q 𝑦 → 𝑥 ∈ 𝐴) ∧ ∃𝑥 ∈ 𝐴 𝑦 <_Q 𝑥))
8	7	r19.21bi 3232	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ 𝐴) → (∀𝑥(𝑥 <_Q 𝑦 → 𝑥 ∈ 𝐴) ∧ ∃𝑥 ∈ 𝐴 𝑦 <_Q 𝑥))
9	8	simprd 496	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝑦 <_Q 𝑥)
10	5, 9	vtoclg 3502	. 2 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 <_Q 𝑥))
11	10	anabsi7 677	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 <_Q 𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 ∀wal 1545 = wceq 1547 ∈ wcel 2119 ∀wral 3054 ∃wrex 3064 Vcvv 3432 ⊊ wpss 3891 ∅c0 4268 class class class wbr 5079 Qcnq 10773 <_Q cltq 10779 Pcnp 10780

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-12 2189 ax-ext 2712

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-sb 2074 df-clab 2719 df-cleq 2732 df-clel 2815 df-ne 2936 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-dif 3893 df-un 3895 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-sn 4563 df-pr 4565 df-op 4569 df-br 5080 df-np 10902

This theorem is referenced by: npomex 10917 prnmadd 10918 genpnmax 10928 1idpr 10950 ltexprlem4 10960 reclem3pr 10970 suplem1pr 10973

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