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Theorem prnmax 10955

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 2817	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
2	1	anbi2d 630	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ P ∧ 𝑦 ∈ 𝐴) ↔ (𝐴 ∈ P ∧ 𝐵 ∈ 𝐴)))
3		breq1 5113	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 <_Q 𝑥 ↔ 𝐵 <_Q 𝑥))
4	3	rexbidv 3158	. . . 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 10948	. . . . . 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 3230	. . . 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 3523	. 2 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 <_Q 𝑥))
11	10	anabsi7 671	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 <_Q 𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∀wal 1538 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 Vcvv 3450 ⊊ wpss 3918 ∅c0 4299 class class class wbr 5110 Qcnq 10812 <_Q cltq 10818 Pcnp 10819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-12 2178 ax-ext 2702

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2709 df-cleq 2722 df-clel 2804 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3920 df-un 3922 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-br 5111 df-np 10941

This theorem is referenced by: npomex 10956 prnmadd 10957 genpnmax 10967 1idpr 10989 ltexprlem4 10999 reclem3pr 11009 suplem1pr 11012

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