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Theorem slerec 27310
Description: A comparison law for surreals considered as cuts of sets of surreals. Definition from [Conway] p. 4. Theorem 4 of [Alling] p. 186. Theorem 2.5 of [Gonshor] p. 9. (Contributed by Scott Fenton, 11-Dec-2021.)
Assertion
Ref Expression
slerec (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 ≤s 𝑌 ↔ (∀𝑑𝐷 𝑋 <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s 𝑌)))
Distinct variable groups:   𝐴,𝑎,𝑑   𝐵,𝑎,𝑑   𝐶,𝑎,𝑑   𝐷,𝑎,𝑑   𝑋,𝑎,𝑑   𝑌,𝑎,𝑑

Proof of Theorem slerec
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scutcl 27293 . . . . . . . 8 (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
21ad3antrrr 729 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐴 |s 𝐵) ∈ No )
3 scutcl 27293 . . . . . . . 8 (𝐶 <<s 𝐷 → (𝐶 |s 𝐷) ∈ No )
43ad3antlr 730 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐶 |s 𝐷) ∈ No )
5 ssltss2 27281 . . . . . . . . 9 (𝐶 <<s 𝐷𝐷 No )
65ad2antlr 726 . . . . . . . 8 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐷 No )
76sselda 3982 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → 𝑑 No )
8 simplr 768 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷))
9 scutcut 27292 . . . . . . . . . . . 12 (𝐶 <<s 𝐷 → ((𝐶 |s 𝐷) ∈ No 𝐶 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐷))
109simp3d 1145 . . . . . . . . . . 11 (𝐶 <<s 𝐷 → {(𝐶 |s 𝐷)} <<s 𝐷)
1110ad2antlr 726 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → {(𝐶 |s 𝐷)} <<s 𝐷)
12 ssltsep 27282 . . . . . . . . . 10 ({(𝐶 |s 𝐷)} <<s 𝐷 → ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑑𝐷 𝑎 <s 𝑑)
1311, 12syl 17 . . . . . . . . 9 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑑𝐷 𝑎 <s 𝑑)
14 ovex 7439 . . . . . . . . . 10 (𝐶 |s 𝐷) ∈ V
15 breq1 5151 . . . . . . . . . . 11 (𝑎 = (𝐶 |s 𝐷) → (𝑎 <s 𝑑 ↔ (𝐶 |s 𝐷) <s 𝑑))
1615ralbidv 3178 . . . . . . . . . 10 (𝑎 = (𝐶 |s 𝐷) → (∀𝑑𝐷 𝑎 <s 𝑑 ↔ ∀𝑑𝐷 (𝐶 |s 𝐷) <s 𝑑))
1714, 16ralsn 4685 . . . . . . . . 9 (∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑑𝐷 𝑎 <s 𝑑 ↔ ∀𝑑𝐷 (𝐶 |s 𝐷) <s 𝑑)
1813, 17sylib 217 . . . . . . . 8 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑑𝐷 (𝐶 |s 𝐷) <s 𝑑)
1918r19.21bi 3249 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐶 |s 𝐷) <s 𝑑)
202, 4, 7, 8, 19slelttrd 27254 . . . . . 6 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐴 |s 𝐵) <s 𝑑)
2120ralrimiva 3147 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑)
22 ssltss1 27280 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐴 No )
2322adantr 482 . . . . . . . . 9 ((𝐴 <<s 𝐵𝐶 <<s 𝐷) → 𝐴 No )
2423adantr 482 . . . . . . . 8 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐴 No )
2524sselda 3982 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → 𝑎 No )
261ad3antrrr 729 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → (𝐴 |s 𝐵) ∈ No )
273ad3antlr 730 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → (𝐶 |s 𝐷) ∈ No )
28 scutcut 27292 . . . . . . . . . . . . 13 (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
2928simp2d 1144 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐴 <<s {(𝐴 |s 𝐵)})
3029adantr 482 . . . . . . . . . . 11 ((𝐴 <<s 𝐵𝐶 <<s 𝐷) → 𝐴 <<s {(𝐴 |s 𝐵)})
3130adantr 482 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐴 <<s {(𝐴 |s 𝐵)})
32 ssltsep 27282 . . . . . . . . . 10 (𝐴 <<s {(𝐴 |s 𝐵)} → ∀𝑎𝐴𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑)
3331, 32syl 17 . . . . . . . . 9 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑎𝐴𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑)
3433r19.21bi 3249 . . . . . . . 8 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → ∀𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑)
35 ovex 7439 . . . . . . . . 9 (𝐴 |s 𝐵) ∈ V
36 breq2 5152 . . . . . . . . 9 (𝑑 = (𝐴 |s 𝐵) → (𝑎 <s 𝑑𝑎 <s (𝐴 |s 𝐵)))
3735, 36ralsn 4685 . . . . . . . 8 (∀𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑𝑎 <s (𝐴 |s 𝐵))
3834, 37sylib 217 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → 𝑎 <s (𝐴 |s 𝐵))
39 simplr 768 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷))
4025, 26, 27, 38, 39sltletrd 27253 . . . . . 6 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → 𝑎 <s (𝐶 |s 𝐷))
4140ralrimiva 3147 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))
4221, 41jca 513 . . . 4 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷)))
43 bdayelon 27268 . . . . . . 7 ( bday ‘(𝐴 |s 𝐵)) ∈ On
4443onordi 6473 . . . . . 6 Ord ( bday ‘(𝐴 |s 𝐵))
45 ordn2lp 6382 . . . . . 6 (Ord ( bday ‘(𝐴 |s 𝐵)) → ¬ (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)) ∧ ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵))))
4644, 45ax-mp 5 . . . . 5 ¬ (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)) ∧ ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵)))
473ad2antlr 726 . . . . . . 7 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) → (𝐶 |s 𝐷) ∈ No )
481adantr 482 . . . . . . . 8 ((𝐴 <<s 𝐵𝐶 <<s 𝐷) → (𝐴 |s 𝐵) ∈ No )
4948adantr 482 . . . . . . 7 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ∈ No )
50 sltnle 27246 . . . . . . 7 (((𝐶 |s 𝐷) ∈ No ∧ (𝐴 |s 𝐵) ∈ No ) → ((𝐶 |s 𝐷) <s (𝐴 |s 𝐵) ↔ ¬ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)))
5147, 49, 50syl2anc 585 . . . . . 6 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) → ((𝐶 |s 𝐷) <s (𝐴 |s 𝐵) ↔ ¬ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)))
523ad3antlr 730 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐶 |s 𝐷) ∈ No )
53 ssltex1 27278 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐴 ∈ V)
5453ad3antrrr 729 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐴 ∈ V)
55 snex 5431 . . . . . . . . . . 11 {(𝐶 |s 𝐷)} ∈ V
5654, 55jctir 522 . . . . . . . . . 10 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐴 ∈ V ∧ {(𝐶 |s 𝐷)} ∈ V))
5722ad3antrrr 729 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐴 No )
5852snssd 4812 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → {(𝐶 |s 𝐷)} ⊆ No )
59 simplrr 777 . . . . . . . . . . . 12 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))
60 breq2 5152 . . . . . . . . . . . . . 14 (𝑑 = (𝐶 |s 𝐷) → (𝑎 <s 𝑑𝑎 <s (𝐶 |s 𝐷)))
6114, 60ralsn 4685 . . . . . . . . . . . . 13 (∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑𝑎 <s (𝐶 |s 𝐷))
6261ralbii 3094 . . . . . . . . . . . 12 (∀𝑎𝐴𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑 ↔ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))
6359, 62sylibr 233 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑎𝐴𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑)
6457, 58, 633jca 1129 . . . . . . . . . 10 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐴 No ∧ {(𝐶 |s 𝐷)} ⊆ No ∧ ∀𝑎𝐴𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑))
65 brsslt 27277 . . . . . . . . . 10 (𝐴 <<s {(𝐶 |s 𝐷)} ↔ ((𝐴 ∈ V ∧ {(𝐶 |s 𝐷)} ∈ V) ∧ (𝐴 No ∧ {(𝐶 |s 𝐷)} ⊆ No ∧ ∀𝑎𝐴𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑)))
6656, 64, 65sylanbrc 584 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐴 <<s {(𝐶 |s 𝐷)})
67 ssltex2 27279 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐵 ∈ V)
6867ad3antrrr 729 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐵 ∈ V)
6968, 55jctil 521 . . . . . . . . . 10 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ({(𝐶 |s 𝐷)} ∈ V ∧ 𝐵 ∈ V))
70 ssltss2 27281 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐵 No )
7170ad3antrrr 729 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐵 No )
7252adantr 482 . . . . . . . . . . . . . 14 (((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏𝐵) → (𝐶 |s 𝐷) ∈ No )
7348ad3antrrr 729 . . . . . . . . . . . . . 14 (((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏𝐵) → (𝐴 |s 𝐵) ∈ No )
7471sselda 3982 . . . . . . . . . . . . . 14 (((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏𝐵) → 𝑏 No )
75 simplr 768 . . . . . . . . . . . . . 14 (((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏𝐵) → (𝐶 |s 𝐷) <s (𝐴 |s 𝐵))
7628simp3d 1145 . . . . . . . . . . . . . . . . . 18 (𝐴 <<s 𝐵 → {(𝐴 |s 𝐵)} <<s 𝐵)
7776ad3antrrr 729 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → {(𝐴 |s 𝐵)} <<s 𝐵)
78 ssltsep 27282 . . . . . . . . . . . . . . . . 17 ({(𝐴 |s 𝐵)} <<s 𝐵 → ∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑏𝐵 𝑎 <s 𝑏)
7977, 78syl 17 . . . . . . . . . . . . . . . 16 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑏𝐵 𝑎 <s 𝑏)
80 breq1 5151 . . . . . . . . . . . . . . . . . 18 (𝑎 = (𝐴 |s 𝐵) → (𝑎 <s 𝑏 ↔ (𝐴 |s 𝐵) <s 𝑏))
8180ralbidv 3178 . . . . . . . . . . . . . . . . 17 (𝑎 = (𝐴 |s 𝐵) → (∀𝑏𝐵 𝑎 <s 𝑏 ↔ ∀𝑏𝐵 (𝐴 |s 𝐵) <s 𝑏))
8235, 81ralsn 4685 . . . . . . . . . . . . . . . 16 (∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑏𝐵 𝑎 <s 𝑏 ↔ ∀𝑏𝐵 (𝐴 |s 𝐵) <s 𝑏)
8379, 82sylib 217 . . . . . . . . . . . . . . 15 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑏𝐵 (𝐴 |s 𝐵) <s 𝑏)
8483r19.21bi 3249 . . . . . . . . . . . . . 14 (((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏𝐵) → (𝐴 |s 𝐵) <s 𝑏)
8572, 73, 74, 75, 84slttrd 27252 . . . . . . . . . . . . 13 (((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏𝐵) → (𝐶 |s 𝐷) <s 𝑏)
8685ralrimiva 3147 . . . . . . . . . . . 12 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑏𝐵 (𝐶 |s 𝐷) <s 𝑏)
87 breq1 5151 . . . . . . . . . . . . . 14 (𝑎 = (𝐶 |s 𝐷) → (𝑎 <s 𝑏 ↔ (𝐶 |s 𝐷) <s 𝑏))
8887ralbidv 3178 . . . . . . . . . . . . 13 (𝑎 = (𝐶 |s 𝐷) → (∀𝑏𝐵 𝑎 <s 𝑏 ↔ ∀𝑏𝐵 (𝐶 |s 𝐷) <s 𝑏))
8914, 88ralsn 4685 . . . . . . . . . . . 12 (∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑏𝐵 𝑎 <s 𝑏 ↔ ∀𝑏𝐵 (𝐶 |s 𝐷) <s 𝑏)
9086, 89sylibr 233 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑏𝐵 𝑎 <s 𝑏)
9158, 71, 903jca 1129 . . . . . . . . . 10 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ({(𝐶 |s 𝐷)} ⊆ No 𝐵 No ∧ ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑏𝐵 𝑎 <s 𝑏))
92 brsslt 27277 . . . . . . . . . 10 ({(𝐶 |s 𝐷)} <<s 𝐵 ↔ (({(𝐶 |s 𝐷)} ∈ V ∧ 𝐵 ∈ V) ∧ ({(𝐶 |s 𝐷)} ⊆ No 𝐵 No ∧ ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑏𝐵 𝑎 <s 𝑏)))
9369, 91, 92sylanbrc 584 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → {(𝐶 |s 𝐷)} <<s 𝐵)
94 sltirr 27239 . . . . . . . . . . . . . 14 ((𝐴 |s 𝐵) ∈ No → ¬ (𝐴 |s 𝐵) <s (𝐴 |s 𝐵))
9549, 94syl 17 . . . . . . . . . . . . 13 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) → ¬ (𝐴 |s 𝐵) <s (𝐴 |s 𝐵))
96 breq1 5151 . . . . . . . . . . . . . 14 ((𝐴 |s 𝐵) = (𝐶 |s 𝐷) → ((𝐴 |s 𝐵) <s (𝐴 |s 𝐵) ↔ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)))
9796notbid 318 . . . . . . . . . . . . 13 ((𝐴 |s 𝐵) = (𝐶 |s 𝐷) → (¬ (𝐴 |s 𝐵) <s (𝐴 |s 𝐵) ↔ ¬ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)))
9895, 97syl5ibcom 244 . . . . . . . . . . . 12 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) → ((𝐴 |s 𝐵) = (𝐶 |s 𝐷) → ¬ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)))
9998necon2ad 2956 . . . . . . . . . . 11 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) → ((𝐶 |s 𝐷) <s (𝐴 |s 𝐵) → (𝐴 |s 𝐵) ≠ (𝐶 |s 𝐷)))
10099imp 408 . . . . . . . . . 10 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐴 |s 𝐵) ≠ (𝐶 |s 𝐷))
101100necomd 2997 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐶 |s 𝐷) ≠ (𝐴 |s 𝐵))
102 scutbdaylt 27309 . . . . . . . . 9 (((𝐶 |s 𝐷) ∈ No ∧ (𝐴 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐵) ∧ (𝐶 |s 𝐷) ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)))
10352, 66, 93, 101, 102syl121anc 1376 . . . . . . . 8 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)))
1041ad3antrrr 729 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐴 |s 𝐵) ∈ No )
105 ssltex1 27278 . . . . . . . . . . . 12 (𝐶 <<s 𝐷𝐶 ∈ V)
106105ad3antlr 730 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐶 ∈ V)
107 snex 5431 . . . . . . . . . . 11 {(𝐴 |s 𝐵)} ∈ V
108106, 107jctir 522 . . . . . . . . . 10 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐶 ∈ V ∧ {(𝐴 |s 𝐵)} ∈ V))
109 ssltss1 27280 . . . . . . . . . . . 12 (𝐶 <<s 𝐷𝐶 No )
110109ad3antlr 730 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐶 No )
111104snssd 4812 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → {(𝐴 |s 𝐵)} ⊆ No )
112110sselda 3982 . . . . . . . . . . . . . 14 (((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐𝐶) → 𝑐 No )
11352adantr 482 . . . . . . . . . . . . . 14 (((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐𝐶) → (𝐶 |s 𝐷) ∈ No )
11448ad3antrrr 729 . . . . . . . . . . . . . 14 (((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐𝐶) → (𝐴 |s 𝐵) ∈ No )
1159simp2d 1144 . . . . . . . . . . . . . . . . . 18 (𝐶 <<s 𝐷𝐶 <<s {(𝐶 |s 𝐷)})
116115ad3antlr 730 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐶 <<s {(𝐶 |s 𝐷)})
117 ssltsep 27282 . . . . . . . . . . . . . . . . 17 (𝐶 <<s {(𝐶 |s 𝐷)} → ∀𝑐𝐶𝑑 ∈ {(𝐶 |s 𝐷)}𝑐 <s 𝑑)
118116, 117syl 17 . . . . . . . . . . . . . . . 16 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑐𝐶𝑑 ∈ {(𝐶 |s 𝐷)}𝑐 <s 𝑑)
119118r19.21bi 3249 . . . . . . . . . . . . . . 15 (((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐𝐶) → ∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑐 <s 𝑑)
120 breq2 5152 . . . . . . . . . . . . . . . 16 (𝑑 = (𝐶 |s 𝐷) → (𝑐 <s 𝑑𝑐 <s (𝐶 |s 𝐷)))
12114, 120ralsn 4685 . . . . . . . . . . . . . . 15 (∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑐 <s 𝑑𝑐 <s (𝐶 |s 𝐷))
122119, 121sylib 217 . . . . . . . . . . . . . 14 (((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐𝐶) → 𝑐 <s (𝐶 |s 𝐷))
123 simplr 768 . . . . . . . . . . . . . 14 (((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐𝐶) → (𝐶 |s 𝐷) <s (𝐴 |s 𝐵))
124112, 113, 114, 122, 123slttrd 27252 . . . . . . . . . . . . 13 (((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐𝐶) → 𝑐 <s (𝐴 |s 𝐵))
125 breq2 5152 . . . . . . . . . . . . . 14 (𝑎 = (𝐴 |s 𝐵) → (𝑐 <s 𝑎𝑐 <s (𝐴 |s 𝐵)))
12635, 125ralsn 4685 . . . . . . . . . . . . 13 (∀𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎𝑐 <s (𝐴 |s 𝐵))
127124, 126sylibr 233 . . . . . . . . . . . 12 (((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐𝐶) → ∀𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎)
128127ralrimiva 3147 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑐𝐶𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎)
129110, 111, 1283jca 1129 . . . . . . . . . 10 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐶 No ∧ {(𝐴 |s 𝐵)} ⊆ No ∧ ∀𝑐𝐶𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎))
130 brsslt 27277 . . . . . . . . . 10 (𝐶 <<s {(𝐴 |s 𝐵)} ↔ ((𝐶 ∈ V ∧ {(𝐴 |s 𝐵)} ∈ V) ∧ (𝐶 No ∧ {(𝐴 |s 𝐵)} ⊆ No ∧ ∀𝑐𝐶𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎)))
131108, 129, 130sylanbrc 584 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐶 <<s {(𝐴 |s 𝐵)})
132 ssltex2 27279 . . . . . . . . . . . 12 (𝐶 <<s 𝐷𝐷 ∈ V)
133132ad3antlr 730 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐷 ∈ V)
134133, 107jctil 521 . . . . . . . . . 10 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ({(𝐴 |s 𝐵)} ∈ V ∧ 𝐷 ∈ V))
1355ad3antlr 730 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐷 No )
136 simplrl 776 . . . . . . . . . . . 12 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑)
137 breq1 5151 . . . . . . . . . . . . . 14 (𝑎 = (𝐴 |s 𝐵) → (𝑎 <s 𝑑 ↔ (𝐴 |s 𝐵) <s 𝑑))
138137ralbidv 3178 . . . . . . . . . . . . 13 (𝑎 = (𝐴 |s 𝐵) → (∀𝑑𝐷 𝑎 <s 𝑑 ↔ ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑))
13935, 138ralsn 4685 . . . . . . . . . . . 12 (∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑑𝐷 𝑎 <s 𝑑 ↔ ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑)
140136, 139sylibr 233 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑑𝐷 𝑎 <s 𝑑)
141111, 135, 1403jca 1129 . . . . . . . . . 10 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ({(𝐴 |s 𝐵)} ⊆ No 𝐷 No ∧ ∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑑𝐷 𝑎 <s 𝑑))
142 brsslt 27277 . . . . . . . . . 10 ({(𝐴 |s 𝐵)} <<s 𝐷 ↔ (({(𝐴 |s 𝐵)} ∈ V ∧ 𝐷 ∈ V) ∧ ({(𝐴 |s 𝐵)} ⊆ No 𝐷 No ∧ ∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑑𝐷 𝑎 <s 𝑑)))
143134, 141, 142sylanbrc 584 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → {(𝐴 |s 𝐵)} <<s 𝐷)
144 scutbdaylt 27309 . . . . . . . . 9 (((𝐴 |s 𝐵) ∈ No ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ (𝐴 |s 𝐵) ≠ (𝐶 |s 𝐷)) → ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵)))
145104, 131, 143, 100, 144syl121anc 1376 . . . . . . . 8 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵)))
146103, 145jca 513 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)) ∧ ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵))))
147146ex 414 . . . . . 6 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) → ((𝐶 |s 𝐷) <s (𝐴 |s 𝐵) → (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)) ∧ ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵)))))
14851, 147sylbird 260 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) → (¬ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷) → (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)) ∧ ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵)))))
14946, 148mt3i 149 . . . 4 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷))
15042, 149impbida 800 . . 3 ((𝐴 <<s 𝐵𝐶 <<s 𝐷) → ((𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷) ↔ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))))
151 breq12 5153 . . . 4 ((𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷)) → (𝑋 ≤s 𝑌 ↔ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)))
152 breq1 5151 . . . . . 6 (𝑋 = (𝐴 |s 𝐵) → (𝑋 <s 𝑑 ↔ (𝐴 |s 𝐵) <s 𝑑))
153152ralbidv 3178 . . . . 5 (𝑋 = (𝐴 |s 𝐵) → (∀𝑑𝐷 𝑋 <s 𝑑 ↔ ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑))
154 breq2 5152 . . . . . 6 (𝑌 = (𝐶 |s 𝐷) → (𝑎 <s 𝑌𝑎 <s (𝐶 |s 𝐷)))
155154ralbidv 3178 . . . . 5 (𝑌 = (𝐶 |s 𝐷) → (∀𝑎𝐴 𝑎 <s 𝑌 ↔ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷)))
156153, 155bi2anan9 638 . . . 4 ((𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷)) → ((∀𝑑𝐷 𝑋 <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s 𝑌) ↔ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))))
157151, 156bibi12d 346 . . 3 ((𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷)) → ((𝑋 ≤s 𝑌 ↔ (∀𝑑𝐷 𝑋 <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s 𝑌)) ↔ ((𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷) ↔ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷)))))
158150, 157imbitrrid 245 . 2 ((𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷)) → ((𝐴 <<s 𝐵𝐶 <<s 𝐷) → (𝑋 ≤s 𝑌 ↔ (∀𝑑𝐷 𝑋 <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s 𝑌))))
159158impcom 409 1 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 ≤s 𝑌 ↔ (∀𝑑𝐷 𝑋 <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  Vcvv 3475  wss 3948  {csn 4628   class class class wbr 5148  Ord word 6361  cfv 6541  (class class class)co 7406   No csur 27133   <s cslt 27134   bday cbday 27135   ≤s csle 27237   <<s csslt 27272   |s cscut 27274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1o 8463  df-2o 8464  df-no 27136  df-slt 27137  df-bday 27138  df-sle 27238  df-sslt 27273  df-scut 27275
This theorem is referenced by:  sltrec  27311
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