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Theorem prnmax 10950

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 2849	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
2	1	anbi2d 639	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ P ∧ 𝑦 ∈ 𝐴) ↔ (𝐴 ∈ P ∧ 𝐵 ∈ 𝐴)))
3		breq1 5102	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 <_Q 𝑥 ↔ 𝐵 <_Q 𝑥))
4	3	rexbidv 3185	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 <_Q 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝐵 <_Q 𝑥))
5	2, 4	imbi12d 346	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ P ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝑦 <_Q 𝑥) ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 <_Q 𝑥)))
6		elnpi 10943	. . . . . 6 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑦 ∈ 𝐴 (∀𝑥(𝑥 <_Q 𝑦 → 𝑥 ∈ 𝐴) ∧ ∃𝑥 ∈ 𝐴 𝑦 <_Q 𝑥)))
7	6	simprbi 501	. . . . 5 ⊢ (𝐴 ∈ P → ∀𝑦 ∈ 𝐴 (∀𝑥(𝑥 <_Q 𝑦 → 𝑥 ∈ 𝐴) ∧ ∃𝑥 ∈ 𝐴 𝑦 <_Q 𝑥))
8	7	r19.21bi 3253	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ 𝐴) → (∀𝑥(𝑥 <_Q 𝑦 → 𝑥 ∈ 𝐴) ∧ ∃𝑥 ∈ 𝐴 𝑦 <_Q 𝑥))
9	8	simprd 499	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝑦 <_Q 𝑥)
10	5, 9	vtoclg 3521	. 2 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 <_Q 𝑥))
11	10	anabsi7 681	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 <_Q 𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1097 ∀wal 1557 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ∃wrex 3085 Vcvv 3453 ⊊ wpss 3905 ∅c0 4285 class class class wbr 5099 Qcnq 10807 <_Q cltq 10813 Pcnp 10814

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-12 2211 ax-ext 2733

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-sb 2090 df-clab 2740 df-cleq 2753 df-clel 2836 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-dif 3907 df-un 3909 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-sn 4582 df-pr 4584 df-op 4588 df-br 5100 df-np 10936

This theorem is referenced by: npomex 10951 prnmadd 10952 genpnmax 10962 1idpr 10984 ltexprlem4 10994 reclem3pr 11004 suplem1pr 11007

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