Theorem prnmax

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
Assertion	prnmax	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ 𝐴 ) → ∃ 𝑥 ∈ 𝐴 𝐵 <Q 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		eleq1	⊢ ( 𝑦 = 𝐵 → ( 𝑦 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴 ) )
2	1	anbi2d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ∈ P ∧ 𝑦 ∈ 𝐴 ) ↔ ( 𝐴 ∈ P ∧ 𝐵 ∈ 𝐴 ) ) )
3		breq1	⊢ ( 𝑦 = 𝐵 → ( 𝑦 <Q 𝑥 ↔ 𝐵 <Q 𝑥 ) )
4	3	rexbidv	⊢ ( 𝑦 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝑦 <Q 𝑥 ↔ ∃ 𝑥 ∈ 𝐴 𝐵 <Q 𝑥 ) )
5	2 4	imbi12d	⊢ ( 𝑦 = 𝐵 → ( ( ( 𝐴 ∈ P ∧ 𝑦 ∈ 𝐴 ) → ∃ 𝑥 ∈ 𝐴 𝑦 <Q 𝑥 ) ↔ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ 𝐴 ) → ∃ 𝑥 ∈ 𝐴 𝐵 <Q 𝑥 ) ) )
6		elnpi	⊢ ( 𝐴 ∈ P ↔ ( ( 𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q ) ∧ ∀ 𝑦 ∈ 𝐴 ( ∀ 𝑥 ( 𝑥 <Q 𝑦 → 𝑥 ∈ 𝐴 ) ∧ ∃ 𝑥 ∈ 𝐴 𝑦 <Q 𝑥 ) ) )
7	6	simprbi	⊢ ( 𝐴 ∈ P → ∀ 𝑦 ∈ 𝐴 ( ∀ 𝑥 ( 𝑥 <Q 𝑦 → 𝑥 ∈ 𝐴 ) ∧ ∃ 𝑥 ∈ 𝐴 𝑦 <Q 𝑥 ) )
8	7	r19.21bi	⊢ ( ( 𝐴 ∈ P ∧ 𝑦 ∈ 𝐴 ) → ( ∀ 𝑥 ( 𝑥 <Q 𝑦 → 𝑥 ∈ 𝐴 ) ∧ ∃ 𝑥 ∈ 𝐴 𝑦 <Q 𝑥 ) )
9	8	simprd	⊢ ( ( 𝐴 ∈ P ∧ 𝑦 ∈ 𝐴 ) → ∃ 𝑥 ∈ 𝐴 𝑦 <Q 𝑥 )
10	5 9	vtoclg	⊢ ( 𝐵 ∈ 𝐴 → ( ( 𝐴 ∈ P ∧ 𝐵 ∈ 𝐴 ) → ∃ 𝑥 ∈ 𝐴 𝐵 <Q 𝑥 ) )
11	10	anabsi7	⊢ ( ( 𝐴 ∈ P ∧ 𝐵 ∈ 𝐴 ) → ∃ 𝑥 ∈ 𝐴 𝐵 <Q 𝑥 )