Theorem cofcut1 27934

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 1209	. . 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 1210	. . 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 1143	. . . . . 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		cutbday 27798	. . . . . 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		sltsex1 27777	. . . . . . . . . . . . 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 733	. . . . . . . . . . 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		sltsss1 27779	. . . . . . . . . . . . 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 733	. . . . . . . . . . 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 4538	. . . . . . . . . 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 1234	. . . . . . . . . . 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 482	. . . . . . . . . 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 486	. . . . . . . . . 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		cofslts 27932	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ 𝐶 <<s {𝑡}) → 𝐴 <<s {𝑡})
17	12, 14, 15, 16	syl3anc 1380	. . . . . . . . 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 414	. . . . . . . 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		sltsex2 27778	. . . . . . . . . . . . 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 733	. . . . . . . . . . 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		sltsss2 27780	. . . . . . . . . . . . 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 733	. . . . . . . . . . 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 4538	. . . . . . . . . 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 1235	. . . . . . . . . . 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 482	. . . . . . . . . 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 486	. . . . . . . . . 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		coinitslts 27933	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝒫 No ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧 ∧ {𝑡} <<s 𝐷) → {𝑡} <<s 𝐵)
30	25, 27, 28, 29	syl3anc 1380	. . . . . . . . 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 414	. . . . . . . 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 616	. . . . . . 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 4009	. . . . . 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 6061	. . . . . 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 4902	. . . . . 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 3951	. . . 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 27763	. . . . . 6 ⊢ bday Fn No
39		ssrab2 4014	. . . . . 6 ⊢ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)} ⊆ No
40		sneq 4568	. . . . . . . . 9 ⊢ (𝑡 = (𝐴 |s 𝐵) → {𝑡} = {(𝐴 |s 𝐵)})
41	40	breq2d 5087	. . . . . . . 8 ⊢ (𝑡 = (𝐴 |s 𝐵) → (𝐶 <<s {𝑡} ↔ 𝐶 <<s {(𝐴 |s 𝐵)}))
42	40	breq1d 5085	. . . . . . . 8 ⊢ (𝑡 = (𝐴 |s 𝐵) → ({𝑡} <<s 𝐷 ↔ {(𝐴 |s 𝐵)} <<s 𝐷))
43	41, 42	anbi12d 639	. . . . . . 7 ⊢ (𝑡 = (𝐴 |s 𝐵) → ((𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷) ↔ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)))
44	3	cutscld 27797	. . . . . . 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 1145	. . . . . . 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 3633	. . . . . 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 7181	. . . . . 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 1474	. . . . 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 4896	. . . . 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 3934	. . 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 7393	. . . . . . 7 ⊢ (𝐴 |s 𝐵) ∈ V
53	52	snnz 4711	. . . . . 6 ⊢ {(𝐴 |s 𝐵)} ≠ ∅
54		sltstr 27801	. . . . . 6 ⊢ ((𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷 ∧ {(𝐴 |s 𝐵)} ≠ ∅) → 𝐶 <<s 𝐷)
55	53, 54	mp3an3 1459	. . . . 5 ⊢ ((𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐶 <<s 𝐷)
56	55	3ad2ant3 1142	. . . 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		eqcuts 27799	. . . 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 591	. . 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 1350	. 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 2747	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 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055  ∃wrex 3065  {crab 3393  Vcvv 3433   ⊆ wss 3885  ∅c0 4264  𝒫 cpw 4532  {csn 4558  ∩ cint 4880   class class class wbr 5075   “ cima 5624   Fn wfn 6484  ‘cfv 6489  (class class class)co 7360   No csur 27625   bday cbday 27627   ≤s cles 27730   <<s cslts 27771   |s ccuts 27773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-int 4881  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1o 8399  df-2o 8400  df-no 27628  df-lts 27629  df-bday 27630  df-les 27731  df-slts 27772  df-cuts 27774

This theorem is referenced by:  cofcut1d  27935  cofcut2  27936