Theorem cofcut1 27239

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 1201	. . 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 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)) → {(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 27143	. . . . . 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 27126	. . . . . . . . . . . . 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 724	. . . . . . . . . . 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 27128	. . . . . . . . . . . . 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 724	. . . . . . . . . . 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 4566	. . . . . . . . . 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 1226	. . . . . . . . . . 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 481	. . . . . . . . . 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 485	. . . . . . . . . 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 27237	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ 𝐶 <<s {𝑡}) → 𝐴 <<s {𝑡})
17	12, 14, 15, 16	syl3anc 1371	. . . . . . . . 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 413	. . . . . . . 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 27127	. . . . . . . . . . . . 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 724	. . . . . . . . . . 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 27129	. . . . . . . . . . . . 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 724	. . . . . . . . . . 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 4566	. . . . . . . . . 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 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 ) → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧)
27	26	adantr 481	. . . . . . . . . 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 485	. . . . . . . . . 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 27238	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝒫 No ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧 ∧ {𝑡} <<s 𝐷) → {𝑡} <<s 𝐵)
30	25, 27, 28, 29	syl3anc 1371	. . . . . . . . 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 413	. . . . . . . 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 4033	. . . . . 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 6054	. . . . . 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 4930	. . . . . 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 3982	. . . 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 27113	. . . . . 6 ⊢ bday Fn No
39		ssrab2 4037	. . . . . 6 ⊢ {𝑡 ∈ No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)} ⊆ No
40		sneq 4596	. . . . . . . . 9 ⊢ (𝑡 = (𝐴 |s 𝐵) → {𝑡} = {(𝐴 |s 𝐵)})
41	40	breq2d 5117	. . . . . . . 8 ⊢ (𝑡 = (𝐴 |s 𝐵) → (𝐶 <<s {𝑡} ↔ 𝐶 <<s {(𝐴 |s 𝐵)}))
42	40	breq1d 5115	. . . . . . . 8 ⊢ (𝑡 = (𝐴 |s 𝐵) → ({𝑡} <<s 𝐷 ↔ {(𝐴 |s 𝐵)} <<s 𝐷))
43	41, 42	anbi12d 631	. . . . . . 7 ⊢ (𝑡 = (𝐴 |s 𝐵) → ((𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷) ↔ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)))
44	3	scutcld 27142	. . . . . . 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 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 7183	. . . . . 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 1465	. . . . 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 4924	. . . . 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 3961	. . 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 7390	. . . . . . 7 ⊢ (𝐴 |s 𝐵) ∈ V
53	52	snnz 4737	. . . . . 6 ⊢ {(𝐴 |s 𝐵)} ≠ ∅
54		sslttr 27146	. . . . . 6 ⊢ ((𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷 ∧ {(𝐴 |s 𝐵)} ≠ ∅) → 𝐶 <<s 𝐷)
55	53, 54	mp3an3 1450	. . . . 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 27144	. . . 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 1342	. 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 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  {crab 3407  Vcvv 3445   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  {csn 4586  ∩ cint 4907   class class class wbr 5105   “ cima 5636   Fn wfn 6491  ‘cfv 6496  (class class class)co 7357   No csur 26988   bday cbday 26990   ≤s csle 27092   <<s csslt 27120   |s cscut 27122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1o 8412  df-2o 8413  df-no 26991  df-slt 26992  df-bday 26993  df-sle 27093  df-sslt 27121  df-scut 27123

This theorem is referenced by:  cofcut1d  27240  cofcut2  27241