Theorem suplocexprlemmu 8085

Description: Lemma for suplocexpr 8092. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-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
suplocexprlemmu	⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝐵))

Distinct variable groups:   𝐴,𝑠,𝑢,𝑤   𝑥,𝐴,𝑦,𝑠,𝑢   𝐵,𝑠   𝜑,𝑠,𝑢,𝑥,𝑦

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

Proof of Theorem suplocexprlemmu

Dummy variables 𝑗 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		suplocexpr.ub	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
2		prop 7842	. . . . . . 7 ⊢ (𝑥 ∈ P → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ P)
3		prmu 7845	. . . . . . 7 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ P → ∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝑥))
4	2, 3	syl 14	. . . . . 6 ⊢ (𝑥 ∈ P → ∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝑥))
5	4	ad2antrl 494	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) → ∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝑥))
6		fo2nd 6392	. . . . . . . . . . . . 13 ⊢ 2nd :V–onto→V
7		fofun 5616	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → Fun 2nd )
8	6, 7	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 2nd
9		fvelima 5754	. . . . . . . . . . . 12 ⊢ ((Fun 2nd ∧ 𝑡 ∈ (2nd “ 𝐴)) → ∃𝑢 ∈ 𝐴 (2nd ‘𝑢) = 𝑡)
10	8, 9	mpan 428	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (2nd “ 𝐴) → ∃𝑢 ∈ 𝐴 (2nd ‘𝑢) = 𝑡)
11	10	adantl 277	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) → ∃𝑢 ∈ 𝐴 (2nd ‘𝑢) = 𝑡)
12		suplocexpr.m	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
13		suplocexpr.loc	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
14	12, 1, 13	suplocexprlemss 8082	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ P)
15	14	ad5antr 500	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝐴 ⊆ P)
16		simprl 535	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝑢 ∈ 𝐴)
17	15, 16	sseldd 3249	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝑢 ∈ P)
18		simprl 535	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) → 𝑥 ∈ P)
19	18	ad4antr 498	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝑥 ∈ P)
20		breq1 4133	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑢 → (𝑦<P 𝑥 ↔ 𝑢<P 𝑥))
21		simprr 537	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) → ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
22	21	ad4antr 498	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
23	20, 22, 16	rspcdva 2934	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝑢<P 𝑥)
24		ltsopr 7963	. . . . . . . . . . . . . . . . 17 ⊢ <P Or P
25		so2nr 4466	. . . . . . . . . . . . . . . . 17 ⊢ ((<P Or P ∧ (𝑢 ∈ P ∧ 𝑥 ∈ P)) → ¬ (𝑢<P 𝑥 ∧ 𝑥<P 𝑢))
26	24, 25	mpan 428	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ P ∧ 𝑥 ∈ P) → ¬ (𝑢<P 𝑥 ∧ 𝑥<P 𝑢))
27	17, 19, 26	syl2anc 415	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → ¬ (𝑢<P 𝑥 ∧ 𝑥<P 𝑢))
28		imnan 701	. . . . . . . . . . . . . . 15 ⊢ ((𝑢<P 𝑥 → ¬ 𝑥<P 𝑢) ↔ ¬ (𝑢<P 𝑥 ∧ 𝑥<P 𝑢))
29	27, 28	sylibr 134	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → (𝑢<P 𝑥 → ¬ 𝑥<P 𝑢))
30	23, 29	mpd 13	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → ¬ 𝑥<P 𝑢)
31		aptiprlemu 8007	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ P ∧ 𝑥 ∈ P ∧ ¬ 𝑥<P 𝑢) → (2nd ‘𝑥) ⊆ (2nd ‘𝑢))
32	17, 19, 30, 31	syl3anc 1278	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → (2nd ‘𝑥) ⊆ (2nd ‘𝑢))
33		simpllr 540	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝑠 ∈ (2nd ‘𝑥))
34	32, 33	sseldd 3249	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝑠 ∈ (2nd ‘𝑢))
35		simprr 537	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → (2nd ‘𝑢) = 𝑡)
36	34, 35	eleqtrd 2317	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ (2nd ‘𝑢) = 𝑡)) → 𝑠 ∈ 𝑡)
37	11, 36	rexlimddv 2673	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) ∧ 𝑡 ∈ (2nd “ 𝐴)) → 𝑠 ∈ 𝑡)
38	37	ralrimiva 2623	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) → ∀𝑡 ∈ (2nd “ 𝐴)𝑠 ∈ 𝑡)
39		vex 2824	. . . . . . . . 9 ⊢ 𝑠 ∈ V
40	39	elint2 3977	. . . . . . . 8 ⊢ (𝑠 ∈ ∩ (2nd “ 𝐴) ↔ ∀𝑡 ∈ (2nd “ 𝐴)𝑠 ∈ 𝑡)
41	38, 40	sylibr 134	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) ∧ 𝑠 ∈ (2nd ‘𝑥)) → 𝑠 ∈ ∩ (2nd “ 𝐴))
42	41	ex 115	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) ∧ 𝑠 ∈ Q) → (𝑠 ∈ (2nd ‘𝑥) → 𝑠 ∈ ∩ (2nd “ 𝐴)))
43	42	reximdva 2652	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) → (∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝑥) → ∃𝑠 ∈ Q 𝑠 ∈ ∩ (2nd “ 𝐴)))
44	5, 43	mpd 13	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ P ∧ ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)) → ∃𝑠 ∈ Q 𝑠 ∈ ∩ (2nd “ 𝐴))
45	1, 44	rexlimddv 2673	. . 3 ⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ ∩ (2nd “ 𝐴))
46		simprr 537	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → 𝑠 ∈ ∩ (2nd “ 𝐴))
47		simprl 535	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → 𝑠 ∈ Q)
48		1nq 7733	. . . . . . . . 9 ⊢ 1Q ∈ Q
49		addclnq 7742	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 1Q ∈ Q) → (𝑠 +Q 1Q) ∈ Q)
50	47, 48, 49	sylancl 417	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → (𝑠 +Q 1Q) ∈ Q)
51		ltaddnq 7774	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 1Q ∈ Q) → 𝑠 <Q (𝑠 +Q 1Q))
52	47, 48, 51	sylancl 417	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → 𝑠 <Q (𝑠 +Q 1Q))
53		breq2 4134	. . . . . . . . 9 ⊢ (𝑗 = (𝑠 +Q 1Q) → (𝑠 <Q 𝑗 ↔ 𝑠 <Q (𝑠 +Q 1Q)))
54	53	rspcev 2929	. . . . . . . 8 ⊢ (((𝑠 +Q 1Q) ∈ Q ∧ 𝑠 <Q (𝑠 +Q 1Q)) → ∃𝑗 ∈ Q 𝑠 <Q 𝑗)
55	50, 52, 54	syl2anc 415	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → ∃𝑗 ∈ Q 𝑠 <Q 𝑗)
56		breq1 4133	. . . . . . . . 9 ⊢ (𝑤 = 𝑠 → (𝑤 <Q 𝑗 ↔ 𝑠 <Q 𝑗))
57	56	rexbidv 2551	. . . . . . . 8 ⊢ (𝑤 = 𝑠 → (∃𝑗 ∈ Q 𝑤 <Q 𝑗 ↔ ∃𝑗 ∈ Q 𝑠 <Q 𝑗))
58	57	rspcev 2929	. . . . . . 7 ⊢ ((𝑠 ∈ ∩ (2nd “ 𝐴) ∧ ∃𝑗 ∈ Q 𝑠 <Q 𝑗) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)∃𝑗 ∈ Q 𝑤 <Q 𝑗)
59	46, 55, 58	syl2anc 415	. . . . . 6 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)∃𝑗 ∈ Q 𝑤 <Q 𝑗)
60		rexcom 2715	. . . . . 6 ⊢ (∃𝑤 ∈ ∩ (2nd “ 𝐴)∃𝑗 ∈ Q 𝑤 <Q 𝑗 ↔ ∃𝑗 ∈ Q ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗)
61	59, 60	sylib 122	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → ∃𝑗 ∈ Q ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗)
62		ssid 3268	. . . . . 6 ⊢ Q ⊆ Q
63		rexss 3315	. . . . . 6 ⊢ (Q ⊆ Q → (∃𝑗 ∈ Q ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗 ↔ ∃𝑗 ∈ Q (𝑗 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗)))
64	62, 63	ax-mp 5	. . . . 5 ⊢ (∃𝑗 ∈ Q ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗 ↔ ∃𝑗 ∈ Q (𝑗 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗))
65	61, 64	sylib 122	. . . 4 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → ∃𝑗 ∈ Q (𝑗 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗))
66		suplocexpr.b	. . . . . . . . . 10 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
67	66	suplocexprlem2b 8081	. . . . . . . . 9 ⊢ (𝐴 ⊆ P → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
68	14, 67	syl 14	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
69	68	eleq2d 2308	. . . . . . 7 ⊢ (𝜑 → (𝑗 ∈ (2nd ‘𝐵) ↔ 𝑗 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
70		breq2 4134	. . . . . . . . 9 ⊢ (𝑢 = 𝑗 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑗))
71	70	rexbidv 2551	. . . . . . . 8 ⊢ (𝑢 = 𝑗 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗))
72	71	elrab 2982	. . . . . . 7 ⊢ (𝑗 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ↔ (𝑗 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗))
73	69, 72	bitrdi 196	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ (2nd ‘𝐵) ↔ (𝑗 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗)))
74	73	rexbidv 2551	. . . . 5 ⊢ (𝜑 → (∃𝑗 ∈ Q 𝑗 ∈ (2nd ‘𝐵) ↔ ∃𝑗 ∈ Q (𝑗 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗)))
75	74	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → (∃𝑗 ∈ Q 𝑗 ∈ (2nd ‘𝐵) ↔ ∃𝑗 ∈ Q (𝑗 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑗)))
76	65, 75	mpbird 167	. . 3 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑠 ∈ ∩ (2nd “ 𝐴))) → ∃𝑗 ∈ Q 𝑗 ∈ (2nd ‘𝐵))
77	45, 76	rexlimddv 2673	. 2 ⊢ (𝜑 → ∃𝑗 ∈ Q 𝑗 ∈ (2nd ‘𝐵))
78		eleq1w 2299	. . 3 ⊢ (𝑗 = 𝑠 → (𝑗 ∈ (2nd ‘𝐵) ↔ 𝑠 ∈ (2nd ‘𝐵)))
79	78	cbvrexv 2787	. 2 ⊢ (∃𝑗 ∈ Q 𝑗 ∈ (2nd ‘𝐵) ↔ ∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝐵))
80	77, 79	sylib 122	1 ⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ (2nd ‘𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532  Vcvv 2821   ⊆ wss 3220  ⟨cop 3712  ∪ cuni 3935  ∩ cint 3970   class class class wbr 4130   Or wor 4440   “ cima 4777  Fun wfun 5371  –onto→wfo 5375  ‘cfv 5377  (class class class)co 6085  1^st c1st 6372  2^nd c2nd 6373  Qcnq 7647  1_Qc1q 7648   +_Q cplq 7649   <_Q cltq 7652  Pcnp 7658  <_P cltp 7662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-iltp 7837

This theorem is used by:  suplocexprlemex  8089