Theorem cofcut1 28225

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 1220	. . 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 1221	. . 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 1154	. . . . . 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 28089	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑡 ∈ No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}))
5	3, 4	syl 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 ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑡 ∈ No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}))
6		sltsex1 28068	. . . . . . . . . . . . 13 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
7	3, 6	syl 18	. . . . . . . . . . . 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 739	. . . . . . . . . . 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 28070	. . . . . . . . . . . . 13 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
10	3, 9	syl 18	. . . . . . . . . . . 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 739	. . . . . . . . . . 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 4563	. . . . . . . . . 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 1245	. . . . . . . . . . 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 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}) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦)
15		simpr 490	. . . . . . . . . 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 28223	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ 𝐶 <<s {𝑡}) → 𝐴 <<s {𝑡})
17	12, 14, 15, 16	syl3anc 1398	. . . . . . . . 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 418	. . . . . . . 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 28069	. . . . . . . . . . . . 13 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
20	3, 19	syl 18	. . . . . . . . . . . 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 739	. . . . . . . . . . 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 28071	. . . . . . . . . . . . 13 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
23	3, 22	syl 18	. . . . . . . . . . . 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 739	. . . . . . . . . . 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 4563	. . . . . . . . . 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 1246	. . . . . . . . . . 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 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) → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧)
28		simpr 490	. . . . . . . . . 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 28224	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝒫 No ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧 ∧ {𝑡} <<s 𝐷) → {𝑡} <<s 𝐵)
30	25, 27, 28, 29	syl3anc 1398	. . . . . . . . 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 418	. . . . . . . 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 621	. . . . . . 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 4023	. . . . . 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 6093	. . . . . 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 4929	. . . . . 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 19	. . . . 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 3965	. . . 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 28053	. . . . . 6 ⊢ bday Fn No
39		ssrab2 4028	. . . . . 6 ⊢ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)} ⊆ No
40		sneq 4594	. . . . . . . . 9 ⊢ (𝑡 = (𝐴 |s 𝐵) → {𝑡} = {(𝐴 |s 𝐵)})
41	40	breq2d 5115	. . . . . . . 8 ⊢ (𝑡 = (𝐴 |s 𝐵) → (𝐶 <<s {𝑡} ↔ 𝐶 <<s {(𝐴 |s 𝐵)}))
42	40	breq1d 5113	. . . . . . . 8 ⊢ (𝑡 = (𝐴 |s 𝐵) → ({𝑡} <<s 𝐷 ↔ {(𝐴 |s 𝐵)} <<s 𝐷))
43	41, 42	anbi12d 644	. . . . . . 7 ⊢ (𝑡 = (𝐴 |s 𝐵) → ((𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷) ↔ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)))
44	3	cutscld 28088	. . . . . . 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 1156	. . . . . . 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 3647	. . . . . 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 7228	. . . . . 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 1494	. . . . 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 4923	. . . . 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 18	. . . 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 3948	. . 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 7442	. . . . . . 7 ⊢ (𝐴 |s 𝐵) ∈ V
53	52	snnz 4737	. . . . . 6 ⊢ {(𝐴 |s 𝐵)} ≠ ∅
54		sltstr 28092	. . . . . 6 ⊢ ((𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷 ∧ {(𝐴 |s 𝐵)} ≠ ∅) → 𝐶 <<s 𝐷)
55	53, 54	mp3an3 1479	. . . . 5 ⊢ ((𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐶 <<s 𝐷)
56	55	3ad2ant3 1153	. . . 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 28090	. . . 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 596	. . 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 1361	. 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 2766	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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ∩ cint 4907   class class class wbr 5103   “ cima 5651   Fn wfn 6523  ‘cfv 6528  (class class class)co 7409   No csur 27916   bday cbday 27918   ≤s cles 28020   <<s cslts 28062   |s ccuts 28064

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-ord 6355  df-on 6356  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-riota 7366  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1o 8455  df-2o 8456  df-no 27919  df-lts 27920  df-bday 27921  df-les 28021  df-slts 28063  df-cuts 28065

This theorem is used by:  cofcut1d  28226  cofcut2  28227