Theorem suplocexprlemloc 8052

Description: Lemma for suplocexpr 8056. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
suplocexpr.ub	⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
suplocexpr.loc	⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
suplocexpr.b	⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩

Assertion

Ref	Expression
suplocexprlemloc	⊢ (𝜑 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑟 ∈ (2nd ‘𝐵))))

Distinct variable groups:   𝑢,𝐴,𝑧,𝑤   𝑥,𝐴,𝑦,𝑢,𝑧   𝑢,𝑞,𝑧,𝑤   𝑥,𝑞,𝑦,𝜑   𝜑,𝑟,𝑤,𝑞   𝜑,𝑧,𝑥,𝑦   𝑢,𝑟

Allowed substitution hints:   𝜑(𝑢)   𝐴(𝑟,𝑞)   𝐵(𝑥,𝑦,𝑧,𝑤,𝑢,𝑟,𝑞)

Proof of Theorem suplocexprlemloc

Dummy variables 𝑠 𝑡 𝑣 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) → 𝑞 <Q 𝑟)
2		ltbtwnnqq 7746	. . . . 5 ⊢ (𝑞 <Q 𝑟 ↔ ∃𝑣 ∈ Q (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))
3	1, 2	sylib 122	. . . 4 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) → ∃𝑣 ∈ Q (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))
4		simplll 535	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) → 𝜑)
5		simprl 531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) → 𝑞 ∈ Q)
6	5	ad2antrr 488	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) → 𝑞 ∈ Q)
7		simprl 531	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) → 𝑣 ∈ Q)
8	4, 6, 7	jca32 310	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) → (𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)))
9		simprrl 541	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) → 𝑞 <Q 𝑣)
10		ltnqpri 7925	. . . . . . . . 9 ⊢ (𝑞 <Q 𝑣 → ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩)
11	10	adantl 277	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) → ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩)
12		breq2 4118	. . . . . . . . . 10 ⊢ (𝑦 = ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩ → (⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑦 ↔ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩))
13		breq2 4118	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩ → (𝑧<P 𝑦 ↔ 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩))
14	13	ralbidv 2544	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩ → (∀𝑧 ∈ 𝐴 𝑧<P 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩))
15	14	orbi2d 798	. . . . . . . . . 10 ⊢ (𝑦 = ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩ → ((∃𝑧 ∈ 𝐴 ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦) ↔ (∃𝑧 ∈ 𝐴 ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩)))
16	12, 15	imbi12d 234	. . . . . . . . 9 ⊢ (𝑦 = ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩ → ((⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑦 → (∃𝑧 ∈ 𝐴 ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)) ↔ (⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩ → (∃𝑧 ∈ 𝐴 ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩))))
17		breq1 4117	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩ → (𝑥<P 𝑦 ↔ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑦))
18		breq1 4117	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩ → (𝑥<P 𝑧 ↔ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧))
19	18	rexbidv 2545	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩ → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ↔ ∃𝑧 ∈ 𝐴 ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧))
20	19	orbi1d 799	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩ → ((∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦) ↔ (∃𝑧 ∈ 𝐴 ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
21	17, 20	imbi12d 234	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩ → ((𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)) ↔ (⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑦 → (∃𝑧 ∈ 𝐴 ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦))))
22	21	ralbidv 2544	. . . . . . . . . 10 ⊢ (𝑥 = ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩ → (∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)) ↔ ∀𝑦 ∈ P (⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑦 → (∃𝑧 ∈ 𝐴 ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦))))
23		suplocexpr.loc	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
24	23	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
25		simplrl 537	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) → 𝑞 ∈ Q)
26		nqprlu 7878	. . . . . . . . . . 11 ⊢ (𝑞 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩ ∈ P)
27	25, 26	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) → ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩ ∈ P)
28	22, 24, 27	rspcdva 2928	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) → ∀𝑦 ∈ P (⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑦 → (∃𝑧 ∈ 𝐴 ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
29		simplrr 538	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) → 𝑣 ∈ Q)
30		nqprlu 7878	. . . . . . . . . 10 ⊢ (𝑣 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩ ∈ P)
31	29, 30	syl 14	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) → ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩ ∈ P)
32	16, 28, 31	rspcdva 2928	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) → (⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩ → (∃𝑧 ∈ 𝐴 ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩)))
33	11, 32	mpd 13	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) → (∃𝑧 ∈ 𝐴 ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩))
34		simpr 110	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) → ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧)
35	27	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) → ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩ ∈ P)
36		suplocexpr.m	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
37		suplocexpr.ub	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
38	36, 37, 23	suplocexprlemss 8046	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ P)
39	38	ad4antr 494	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) → 𝐴 ⊆ P)
40		simplr 529	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) → 𝑧 ∈ 𝐴)
41	39, 40	sseldd 3243	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) → 𝑧 ∈ P)
42		ltdfpr 7837	. . . . . . . . . . . . 13 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩ ∈ P ∧ 𝑧 ∈ P) → (⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧 ↔ ∃𝑤 ∈ Q (𝑤 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩) ∧ 𝑤 ∈ (1st ‘𝑧))))
43	35, 41, 42	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) → (⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧 ↔ ∃𝑤 ∈ Q (𝑤 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩) ∧ 𝑤 ∈ (1st ‘𝑧))))
44	34, 43	mpbid 147	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) → ∃𝑤 ∈ Q (𝑤 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩) ∧ 𝑤 ∈ (1st ‘𝑧)))
45		vex 2818	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
46		breq2 4118	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑤 → (𝑞 <Q 𝑢 ↔ 𝑞 <Q 𝑤))
47		ltnqex 7880	. . . . . . . . . . . . . . 15 ⊢ {𝑙 ∣ 𝑙 <Q 𝑞} ∈ V
48		gtnqex 7881	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∣ 𝑞 <Q 𝑢} ∈ V
49	47, 48	op2nd 6354	. . . . . . . . . . . . . 14 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩) = {𝑢 ∣ 𝑞 <Q 𝑢}
50	45, 46, 49	elab2 2968	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩) ↔ 𝑞 <Q 𝑤)
51	50	anbi1i 458	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩) ∧ 𝑤 ∈ (1st ‘𝑧)) ↔ (𝑞 <Q 𝑤 ∧ 𝑤 ∈ (1st ‘𝑧)))
52	51	rexbii 2551	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ Q (𝑤 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩) ∧ 𝑤 ∈ (1st ‘𝑧)) ↔ ∃𝑤 ∈ Q (𝑞 <Q 𝑤 ∧ 𝑤 ∈ (1st ‘𝑧)))
53	44, 52	sylib 122	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) → ∃𝑤 ∈ Q (𝑞 <Q 𝑤 ∧ 𝑤 ∈ (1st ‘𝑧)))
54		simpllr 536	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤 ∈ Q ∧ (𝑞 <Q 𝑤 ∧ 𝑤 ∈ (1st ‘𝑧)))) → 𝑧 ∈ 𝐴)
55		simprrl 541	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤 ∈ Q ∧ (𝑞 <Q 𝑤 ∧ 𝑤 ∈ (1st ‘𝑧)))) → 𝑞 <Q 𝑤)
56	41	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤 ∈ Q ∧ (𝑞 <Q 𝑤 ∧ 𝑤 ∈ (1st ‘𝑧)))) → 𝑧 ∈ P)
57		prop 7806	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ P → ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ P)
58	56, 57	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤 ∈ Q ∧ (𝑞 <Q 𝑤 ∧ 𝑤 ∈ (1st ‘𝑧)))) → ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ P)
59		simprrr 542	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤 ∈ Q ∧ (𝑞 <Q 𝑤 ∧ 𝑤 ∈ (1st ‘𝑧)))) → 𝑤 ∈ (1st ‘𝑧))
60		prcdnql 7815	. . . . . . . . . . . . . . . 16 ⊢ ((⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ P ∧ 𝑤 ∈ (1st ‘𝑧)) → (𝑞 <Q 𝑤 → 𝑞 ∈ (1st ‘𝑧)))
61	58, 59, 60	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤 ∈ Q ∧ (𝑞 <Q 𝑤 ∧ 𝑤 ∈ (1st ‘𝑧)))) → (𝑞 <Q 𝑤 → 𝑞 ∈ (1st ‘𝑧)))
62	55, 61	mpd 13	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤 ∈ Q ∧ (𝑞 <Q 𝑤 ∧ 𝑤 ∈ (1st ‘𝑧)))) → 𝑞 ∈ (1st ‘𝑧))
63	54, 62	jca 306	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤 ∈ Q ∧ (𝑞 <Q 𝑤 ∧ 𝑤 ∈ (1st ‘𝑧)))) → (𝑧 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑧)))
64	63	19.8ad 1640	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤 ∈ Q ∧ (𝑞 <Q 𝑤 ∧ 𝑤 ∈ (1st ‘𝑧)))) → ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑧)))
65		df-rex 2528	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ 𝐴 𝑞 ∈ (1st ‘𝑧) ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑧)))
66	64, 65	sylibr 134	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤 ∈ Q ∧ (𝑞 <Q 𝑤 ∧ 𝑤 ∈ (1st ‘𝑧)))) → ∃𝑧 ∈ 𝐴 𝑞 ∈ (1st ‘𝑧))
67		suplocexprlemell 8044	. . . . . . . . . . 11 ⊢ (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 𝑞 ∈ (1st ‘𝑧))
68	66, 67	sylibr 134	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤 ∈ Q ∧ (𝑞 <Q 𝑤 ∧ 𝑤 ∈ (1st ‘𝑧)))) → 𝑞 ∈ ∪ (1st “ 𝐴))
69	53, 68	rexlimddv 2667	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧 ∈ 𝐴) ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧) → 𝑞 ∈ ∪ (1st “ 𝐴))
70	69	rexlimdva2 2665	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) → (∃𝑧 ∈ 𝐴 ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧 → 𝑞 ∈ ∪ (1st “ 𝐴)))
71		fo2nd 6365	. . . . . . . . . . . . . . 15 ⊢ 2nd :V–onto→V
72		fofun 5596	. . . . . . . . . . . . . . 15 ⊢ (2nd :V–onto→V → Fun 2nd )
73	71, 72	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ Fun 2nd
74		fvelima 5733	. . . . . . . . . . . . . 14 ⊢ ((Fun 2nd ∧ 𝑠 ∈ (2nd “ 𝐴)) → ∃𝑡 ∈ 𝐴 (2nd ‘𝑡) = 𝑠)
75	73, 74	mpan 424	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (2nd “ 𝐴) → ∃𝑡 ∈ 𝐴 (2nd ‘𝑡) = 𝑠)
76	75	adantl 277	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd “ 𝐴)) → ∃𝑡 ∈ 𝐴 (2nd ‘𝑡) = 𝑠)
77		breq1 4117	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑡 → (𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩ ↔ 𝑡<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩))
78		simpllr 536	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ (2nd ‘𝑡) = 𝑠)) → ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩)
79		simprl 531	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ (2nd ‘𝑡) = 𝑠)) → 𝑡 ∈ 𝐴)
80	77, 78, 79	rspcdva 2928	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ (2nd ‘𝑡) = 𝑠)) → 𝑡<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩)
81	29	ad3antrrr 492	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ (2nd ‘𝑡) = 𝑠)) → 𝑣 ∈ Q)
82	38	ad5antr 496	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ (2nd ‘𝑡) = 𝑠)) → 𝐴 ⊆ P)
83	82, 79	sseldd 3243	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ (2nd ‘𝑡) = 𝑠)) → 𝑡 ∈ P)
84		nqpru 7883	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ Q ∧ 𝑡 ∈ P) → (𝑣 ∈ (2nd ‘𝑡) ↔ 𝑡<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩))
85	81, 83, 84	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ (2nd ‘𝑡) = 𝑠)) → (𝑣 ∈ (2nd ‘𝑡) ↔ 𝑡<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩))
86	80, 85	mpbird 167	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ (2nd ‘𝑡) = 𝑠)) → 𝑣 ∈ (2nd ‘𝑡))
87		simprr 533	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ (2nd ‘𝑡) = 𝑠)) → (2nd ‘𝑡) = 𝑠)
88	86, 87	eleqtrd 2313	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ (2nd ‘𝑡) = 𝑠)) → 𝑣 ∈ 𝑠)
89	76, 88	rexlimddv 2667	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd “ 𝐴)) → 𝑣 ∈ 𝑠)
90	89	ralrimiva 2617	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩) → ∀𝑠 ∈ (2nd “ 𝐴)𝑣 ∈ 𝑠)
91		vex 2818	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
92	91	elint2 3961	. . . . . . . . . 10 ⊢ (𝑣 ∈ ∩ (2nd “ 𝐴) ↔ ∀𝑠 ∈ (2nd “ 𝐴)𝑣 ∈ 𝑠)
93	90, 92	sylibr 134	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩) → 𝑣 ∈ ∩ (2nd “ 𝐴))
94	93	ex 115	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) → (∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩ → 𝑣 ∈ ∩ (2nd “ 𝐴)))
95	70, 94	orim12d 794	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) → ((∃𝑧 ∈ 𝐴 ⟨{𝑙 ∣ 𝑙 <Q 𝑞}, {𝑢 ∣ 𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P ⟨{𝑙 ∣ 𝑙 <Q 𝑣}, {𝑢 ∣ 𝑣 <Q 𝑢}⟩) → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑣 ∈ ∩ (2nd “ 𝐴))))
96	33, 95	mpd 13	. . . . . 6 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑣 ∈ Q)) ∧ 𝑞 <Q 𝑣) → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑣 ∈ ∩ (2nd “ 𝐴)))
97	8, 9, 96	syl2anc 411	. . . . 5 ⊢ ((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑣 ∈ ∩ (2nd “ 𝐴)))
98		breq2 4118	. . . . . . . . . 10 ⊢ (𝑢 = 𝑟 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑟))
99	98	rexbidv 2545	. . . . . . . . 9 ⊢ (𝑢 = 𝑟 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑟))
100		simprr 533	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) → 𝑟 ∈ Q)
101	100	ad3antrrr 492	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) ∧ 𝑣 ∈ ∩ (2nd “ 𝐴)) → 𝑟 ∈ Q)
102		simpr 110	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) ∧ 𝑣 ∈ ∩ (2nd “ 𝐴)) → 𝑣 ∈ ∩ (2nd “ 𝐴))
103		simprrr 542	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) → 𝑣 <Q 𝑟)
104	103	adantr 276	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) ∧ 𝑣 ∈ ∩ (2nd “ 𝐴)) → 𝑣 <Q 𝑟)
105		breq1 4117	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → (𝑤 <Q 𝑟 ↔ 𝑣 <Q 𝑟))
106	105	rspcev 2923	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ∩ (2nd “ 𝐴) ∧ 𝑣 <Q 𝑟) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑟)
107	102, 104, 106	syl2anc 411	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) ∧ 𝑣 ∈ ∩ (2nd “ 𝐴)) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑟)
108	99, 101, 107	elrabd 2978	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) ∧ 𝑣 ∈ ∩ (2nd “ 𝐴)) → 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
109		suplocexpr.b	. . . . . . . . . . . 12 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
110	109	suplocexprlem2b 8045	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ P → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
111	38, 110	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
112	111	eleq2d 2304	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 ∈ (2nd ‘𝐵) ↔ 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
113	112	ad4antr 494	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) ∧ 𝑣 ∈ ∩ (2nd “ 𝐴)) → (𝑟 ∈ (2nd ‘𝐵) ↔ 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
114	108, 113	mpbird 167	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) ∧ 𝑣 ∈ ∩ (2nd “ 𝐴)) → 𝑟 ∈ (2nd ‘𝐵))
115	114	ex 115	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) → (𝑣 ∈ ∩ (2nd “ 𝐴) → 𝑟 ∈ (2nd ‘𝐵)))
116	115	orim2d 796	. . . . 5 ⊢ ((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) → ((𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑣 ∈ ∩ (2nd “ 𝐴)) → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑟 ∈ (2nd ‘𝐵))))
117	97, 116	mpd 13	. . . 4 ⊢ ((((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣 ∈ Q ∧ (𝑞 <Q 𝑣 ∧ 𝑣 <Q 𝑟))) → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑟 ∈ (2nd ‘𝐵)))
118	3, 117	rexlimddv 2667	. . 3 ⊢ (((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑟 ∈ (2nd ‘𝐵)))
119	118	ex 115	. 2 ⊢ ((𝜑 ∧ (𝑞 ∈ Q ∧ 𝑟 ∈ Q)) → (𝑞 <Q 𝑟 → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑟 ∈ (2nd ‘𝐵))))
120	119	ralrimivva 2626	1 ⊢ (𝜑 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑟 ∈ (2nd ‘𝐵))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2205  {cab 2220  ∀wral 2522  ∃wrex 2523  {crab 2526  Vcvv 2815   ⊆ wss 3214  ⟨cop 3697  ∪ cuni 3919  ∩ cint 3954   class class class wbr 4114   “ cima 4757  Fun wfun 5351  –onto→wfo 5355  ‘cfv 5357  1^st c1st 6345  2^nd c2nd 6346  Qcnq 7611   <_Q cltq 7616  Pcnp 7622  <_P cltp 7626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-inp 7797  df-iltp 7801

This theorem is referenced by:  suplocexprlemex  8053