Theorem coinitsslt

Description: If B is coinitial with C and A precedes C , then A precedes B . (Contributed by Scott Fenton, 24-Sep-2024)

	Ref	Expression
Assertion	coinitsslt	⊢ ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) → 𝐴 <<s 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ssltex1	⊢ ( 𝐴 <<s 𝐶 → 𝐴 ∈ V )
2	1	3ad2ant3	⊢ ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) → 𝐴 ∈ V )
3		simp1	⊢ ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) → 𝐵 ∈ 𝒫 No )
4		ssltss1	⊢ ( 𝐴 <<s 𝐶 → 𝐴 ⊆ No )
5	4	3ad2ant3	⊢ ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) → 𝐴 ⊆ No )
6	3	elpwid	⊢ ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) → 𝐵 ⊆ No )
7		breq2	⊢ ( 𝑥 = 𝑏 → ( 𝑦 ≤s 𝑥 ↔ 𝑦 ≤s 𝑏 ) )
8	7	rexbidv	⊢ ( 𝑥 = 𝑏 → ( ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ↔ ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑏 ) )
9		simp12	⊢ ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) → ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 )
10		simp3	⊢ ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) → 𝑏 ∈ 𝐵 )
11	8 9 10	rspcdva	⊢ ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑏 )
12		breq1	⊢ ( 𝑦 = 𝑐 → ( 𝑦 ≤s 𝑏 ↔ 𝑐 ≤s 𝑏 ) )
13	12	cbvrexvw	⊢ ( ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑏 ↔ ∃ 𝑐 ∈ 𝐶 𝑐 ≤s 𝑏 )
14	11 13	sylib	⊢ ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) → ∃ 𝑐 ∈ 𝐶 𝑐 ≤s 𝑏 )
15		simpl13	⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝐴 <<s 𝐶 )
16	15 4	syl	⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝐴 ⊆ No )
17		simpl2	⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑎 ∈ 𝐴 )
18	16 17	sseldd	⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑎 ∈ No )
19		ssltss2	⊢ ( 𝐴 <<s 𝐶 → 𝐶 ⊆ No )
20	15 19	syl	⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝐶 ⊆ No )
21		simprl	⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑐 ∈ 𝐶 )
22	20 21	sseldd	⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑐 ∈ No )
23		simpl1	⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) )
24	23 6	syl	⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝐵 ⊆ No )
25		simpl3	⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑏 ∈ 𝐵 )
26	24 25	sseldd	⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑏 ∈ No )
27	15 17 21	ssltsepcd	⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑎 <s 𝑐 )
28		simprr	⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑐 ≤s 𝑏 )
29	18 22 26 27 28	sltletrd	⊢ ( ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ∧ ( 𝑐 ∈ 𝐶 ∧ 𝑐 ≤s 𝑏 ) ) → 𝑎 <s 𝑏 )
30	14 29	rexlimddv	⊢ ( ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) → 𝑎 <s 𝑏 )
31	2 3 5 6 30	ssltd	⊢ ( ( 𝐵 ∈ 𝒫 No ∧ ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶 ) → 𝐴 <<s 𝐵 )