Theorem cofcut1 27885

Description: If 𝐶 is cofinal with 𝐴 and 𝐷 is coinitial with 𝐵 and the cut of 𝐴 and 𝐵 lies between 𝐶 and 𝐷, then the cut of 𝐶 and 𝐷 is equal to the cut of 𝐴 and 𝐵. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 25-Sep-2024.)

Assertion

Ref	Expression
cofcut1	⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))

Distinct variable groups:   𝑥,𝐴   𝑧,𝐵   𝑥,𝐶,𝑦   𝑤,𝐷,𝑧

Allowed substitution hints:   𝐴(𝑦,𝑧,𝑤)   𝐵(𝑥,𝑦,𝑤)   𝐶(𝑧,𝑤)   𝐷(𝑥,𝑦)

Proof of Theorem cofcut1

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3l 1202	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐶 <<s {(𝐴 |s 𝐵)})
2		simp3r 1203	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → {(𝐴 |s 𝐵)} <<s 𝐷)
3		simp1 1136	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐴 <<s 𝐵)
4		scutbday 27773	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑡 ∈ No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}))
5	3, 4	syl 17	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑡 ∈ No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}))
6		ssltex1 27755	. . . . . . . . . . . . 13 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
7	3, 6	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐴 ∈ V)
8	7	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) ∧ 𝐶 <<s {𝑡}) → 𝐴 ∈ V)
9		ssltss1 27757	. . . . . . . . . . . . 13 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
10	3, 9	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐴 ⊆ No )
11	10	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) ∧ 𝐶 <<s {𝑡}) → 𝐴 ⊆ No )
12	8, 11	elpwd 4586	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) ∧ 𝐶 <<s {𝑡}) → 𝐴 ∈ 𝒫 No )
13		simpl2l 1227	. . . . . . . . . . 11 ⊢ (((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦)
14	13	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) ∧ 𝐶 <<s {𝑡}) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦)
15		simpr 484	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) ∧ 𝐶 <<s {𝑡}) → 𝐶 <<s {𝑡})
16		cofsslt 27883	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ 𝐶 <<s {𝑡}) → 𝐴 <<s {𝑡})
17	12, 14, 15, 16	syl3anc 1373	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) ∧ 𝐶 <<s {𝑡}) → 𝐴 <<s {𝑡})
18	17	ex 412	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) → (𝐶 <<s {𝑡} → 𝐴 <<s {𝑡}))
19		ssltex2 27756	. . . . . . . . . . . . 13 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
20	3, 19	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐵 ∈ V)
21	20	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) ∧ {𝑡} <<s 𝐷) → 𝐵 ∈ V)
22		ssltss2 27758	. . . . . . . . . . . . 13 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
23	3, 22	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐵 ⊆ No )
24	23	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) ∧ {𝑡} <<s 𝐷) → 𝐵 ⊆ No )
25	21, 24	elpwd 4586	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) ∧ {𝑡} <<s 𝐷) → 𝐵 ∈ 𝒫 No )
26		simpl2r 1228	. . . . . . . . . . 11 ⊢ (((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧)
27	26	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) ∧ {𝑡} <<s 𝐷) → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧)
28		simpr 484	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) ∧ {𝑡} <<s 𝐷) → {𝑡} <<s 𝐷)
29		coinitsslt 27884	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝒫 No ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧 ∧ {𝑡} <<s 𝐷) → {𝑡} <<s 𝐵)
30	25, 27, 28, 29	syl3anc 1373	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) ∧ {𝑡} <<s 𝐷) → {𝑡} <<s 𝐵)
31	30	ex 412	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) → ({𝑡} <<s 𝐷 → {𝑡} <<s 𝐵))
32	18, 31	anim12d 609	. . . . . . 7 ⊢ (((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈ No ) → ((𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷) → (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)))
33	32	ss2rabdv 4056	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)} ⊆ {𝑡 ∈ No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)})
34		imass2 6094	. . . . . 6 ⊢ ({𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)} ⊆ {𝑡 ∈ No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)} → ( bday “ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}) ⊆ ( bday “ {𝑡 ∈ No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}))
35		intss 4950	. . . . . 6 ⊢ (( bday “ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}) ⊆ ( bday “ {𝑡 ∈ No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}) → ∩ ( bday “ {𝑡 ∈ No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}) ⊆ ∩ ( bday “ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}))
36	33, 34, 35	3syl 18	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → ∩ ( bday “ {𝑡 ∈ No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}) ⊆ ∩ ( bday “ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}))
37	5, 36	eqsstrd 3998	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ∩ ( bday “ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}))
38		bdayfn 27742	. . . . . 6 ⊢ bday Fn No
39		ssrab2 4060	. . . . . 6 ⊢ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)} ⊆ No
40		sneq 4616	. . . . . . . . 9 ⊢ (𝑡 = (𝐴 |s 𝐵) → {𝑡} = {(𝐴 |s 𝐵)})
41	40	breq2d 5136	. . . . . . . 8 ⊢ (𝑡 = (𝐴 |s 𝐵) → (𝐶 <<s {𝑡} ↔ 𝐶 <<s {(𝐴 |s 𝐵)}))
42	40	breq1d 5134	. . . . . . . 8 ⊢ (𝑡 = (𝐴 |s 𝐵) → ({𝑡} <<s 𝐷 ↔ {(𝐴 |s 𝐵)} <<s 𝐷))
43	41, 42	anbi12d 632	. . . . . . 7 ⊢ (𝑡 = (𝐴 |s 𝐵) → ((𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷) ↔ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)))
44	3	scutcld 27772	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) ∈ No )
45		simp3 1138	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷))
46	43, 44, 45	elrabd 3678	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) ∈ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)})
47		fnfvima 7230	. . . . . 6 ⊢ (( bday Fn No ∧ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)} ⊆ No ∧ (𝐴 |s 𝐵) ∈ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday “ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}))
48	38, 39, 46, 47	mp3an12i 1467	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday “ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}))
49		intss1 4944	. . . . 5 ⊢ (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday “ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}) → ∩ ( bday “ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}) ⊆ ( bday ‘(𝐴 |s 𝐵)))
50	48, 49	syl 17	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → ∩ ( bday “ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}) ⊆ ( bday ‘(𝐴 |s 𝐵)))
51	37, 50	eqssd 3981	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}))
52		ovex 7443	. . . . . . 7 ⊢ (𝐴 |s 𝐵) ∈ V
53	52	snnz 4757	. . . . . 6 ⊢ {(𝐴 |s 𝐵)} ≠ ∅
54		sslttr 27776	. . . . . 6 ⊢ ((𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷 ∧ {(𝐴 |s 𝐵)} ≠ ∅) → 𝐶 <<s 𝐷)
55	53, 54	mp3an3 1452	. . . . 5 ⊢ ((𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐶 <<s 𝐷)
56	55	3ad2ant3 1135	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐶 <<s 𝐷)
57		eqscut 27774	. . . 4 ⊢ ((𝐶 <<s 𝐷 ∧ (𝐴 |s 𝐵) ∈ No ) → ((𝐶 |s 𝐷) = (𝐴 |s 𝐵) ↔ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷 ∧ ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}))))
58	56, 44, 57	syl2anc 584	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → ((𝐶 |s 𝐷) = (𝐴 |s 𝐵) ↔ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷 ∧ ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}))))
59	1, 2, 51, 58	mpbir3and 1343	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐶 |s 𝐷) = (𝐴 |s 𝐵))
60	59	eqcomd 2742	1 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  {crab 3420  Vcvv 3464   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580  {csn 4606  ∩ cint 4927   class class class wbr 5124   “ cima 5662   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410   No csur 27608   bday cbday 27610   ≤s csle 27713   <<s csslt 27749   |s cscut 27751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1o 8485  df-2o 8486  df-no 27611  df-slt 27612  df-bday 27613  df-sle 27714  df-sslt 27750  df-scut 27752

This theorem is referenced by:  cofcut1d  27886  cofcut2  27887