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Theorem cofcut1 27969
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 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 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.
StepHypRef Expression
1 simp3l 1200 . . 3 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐶 <<s {(𝐴 |s 𝐵)})
2 simp3r 1201 . . 3 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → {(𝐴 |s 𝐵)} <<s 𝐷)
3 simp1 1135 . . . . . 6 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐴 <<s 𝐵)
4 scutbday 27864 . . . . . 6 (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}))
53, 4syl 17 . . . . 5 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 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 27846 . . . . . . . . . . . . 13 (𝐴 <<s 𝐵𝐴 ∈ V)
73, 6syl 17 . . . . . . . . . . . 12 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐴 ∈ V)
87ad2antrr 726 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) ∧ 𝐶 <<s {𝑡}) → 𝐴 ∈ V)
9 ssltss1 27848 . . . . . . . . . . . . 13 (𝐴 <<s 𝐵𝐴 No )
103, 9syl 17 . . . . . . . . . . . 12 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐴 No )
1110ad2antrr 726 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) ∧ 𝐶 <<s {𝑡}) → 𝐴 No )
128, 11elpwd 4611 . . . . . . . . . 10 ((((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) ∧ 𝐶 <<s {𝑡}) → 𝐴 ∈ 𝒫 No )
13 simpl2l 1225 . . . . . . . . . . 11 (((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) → ∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦)
1413adantr 480 . . . . . . . . . 10 ((((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) ∧ 𝐶 <<s {𝑡}) → ∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦)
15 simpr 484 . . . . . . . . . 10 ((((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) ∧ 𝐶 <<s {𝑡}) → 𝐶 <<s {𝑡})
16 cofsslt 27967 . . . . . . . . . 10 ((𝐴 ∈ 𝒫 No ∧ ∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦𝐶 <<s {𝑡}) → 𝐴 <<s {𝑡})
1712, 14, 15, 16syl3anc 1370 . . . . . . . . 9 ((((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) ∧ 𝐶 <<s {𝑡}) → 𝐴 <<s {𝑡})
1817ex 412 . . . . . . . 8 (((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) → (𝐶 <<s {𝑡} → 𝐴 <<s {𝑡}))
19 ssltex2 27847 . . . . . . . . . . . . 13 (𝐴 <<s 𝐵𝐵 ∈ V)
203, 19syl 17 . . . . . . . . . . . 12 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐵 ∈ V)
2120ad2antrr 726 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) ∧ {𝑡} <<s 𝐷) → 𝐵 ∈ V)
22 ssltss2 27849 . . . . . . . . . . . . 13 (𝐴 <<s 𝐵𝐵 No )
233, 22syl 17 . . . . . . . . . . . 12 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐵 No )
2423ad2antrr 726 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) ∧ {𝑡} <<s 𝐷) → 𝐵 No )
2521, 24elpwd 4611 . . . . . . . . . 10 ((((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) ∧ {𝑡} <<s 𝐷) → 𝐵 ∈ 𝒫 No )
26 simpl2r 1226 . . . . . . . . . . 11 (((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) → ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧)
2726adantr 480 . . . . . . . . . 10 ((((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) ∧ {𝑡} <<s 𝐷) → ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧)
28 simpr 484 . . . . . . . . . 10 ((((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) ∧ {𝑡} <<s 𝐷) → {𝑡} <<s 𝐷)
29 coinitsslt 27968 . . . . . . . . . 10 ((𝐵 ∈ 𝒫 No ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧 ∧ {𝑡} <<s 𝐷) → {𝑡} <<s 𝐵)
3025, 27, 28, 29syl3anc 1370 . . . . . . . . 9 ((((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) ∧ {𝑡} <<s 𝐷) → {𝑡} <<s 𝐵)
3130ex 412 . . . . . . . 8 (((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 No ) → ({𝑡} <<s 𝐷 → {𝑡} <<s 𝐵))
3218, 31anim12d 609 . . . . . . 7 (((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 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)))
3332ss2rabdv 4086 . . . . . 6 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 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 6123 . . . . . 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 4974 . . . . . 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)}))
3633, 34, 353syl 18 . . . . 5 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 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)}))
375, 36eqsstrd 4034 . . . 4 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 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 27833 . . . . . 6 bday Fn No
39 ssrab2 4090 . . . . . 6 {𝑡 No ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)} ⊆ No
40 sneq 4641 . . . . . . . . 9 (𝑡 = (𝐴 |s 𝐵) → {𝑡} = {(𝐴 |s 𝐵)})
4140breq2d 5160 . . . . . . . 8 (𝑡 = (𝐴 |s 𝐵) → (𝐶 <<s {𝑡} ↔ 𝐶 <<s {(𝐴 |s 𝐵)}))
4240breq1d 5158 . . . . . . . 8 (𝑡 = (𝐴 |s 𝐵) → ({𝑡} <<s 𝐷 ↔ {(𝐴 |s 𝐵)} <<s 𝐷))
4341, 42anbi12d 632 . . . . . . 7 (𝑡 = (𝐴 |s 𝐵) → ((𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷) ↔ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)))
443scutcld 27863 . . . . . . 7 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) ∈ No )
45 simp3 1137 . . . . . . 7 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷))
4643, 44, 45elrabd 3697 . . . . . 6 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 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 7253 . . . . . 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)}))
4838, 39, 46, 47mp3an12i 1464 . . . . 5 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 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 4968 . . . . 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)))
5048, 49syl 17 . . . 4 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 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)))
5137, 50eqssd 4013 . . 3 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 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 7464 . . . . . . 7 (𝐴 |s 𝐵) ∈ V
5352snnz 4781 . . . . . 6 {(𝐴 |s 𝐵)} ≠ ∅
54 sslttr 27867 . . . . . 6 ((𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷 ∧ {(𝐴 |s 𝐵)} ≠ ∅) → 𝐶 <<s 𝐷)
5553, 54mp3an3 1449 . . . . 5 ((𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐶 <<s 𝐷)
56553ad2ant3 1134 . . . 4 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐶 <<s 𝐷)
57 eqscut 27865 . . . 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)}))))
5856, 44, 57syl2anc 584 . . 3 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 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)}))))
591, 2, 51, 58mpbir3and 1341 . 2 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐶 |s 𝐷) = (𝐴 |s 𝐵))
6059eqcomd 2741 1 ((𝐴 <<s 𝐵 ∧ (∀𝑥𝐴𝑦𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧𝐵𝑤𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   cint 4951   class class class wbr 5148  cima 5692   Fn wfn 6558  cfv 6563  (class class class)co 7431   No csur 27699   bday cbday 27701   ≤s csle 27804   <<s csslt 27840   |s cscut 27842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843
This theorem is referenced by:  cofcut1d  27970  cofcut2  27971
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