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Theorem slerec 27158
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 27141 . . . . . . . 8 (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
21ad3antrrr 728 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐴 |s 𝐵) ∈ No )
3 scutcl 27141 . . . . . . . 8 (𝐶 <<s 𝐷 → (𝐶 |s 𝐷) ∈ No )
43ad3antlr 729 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐶 |s 𝐷) ∈ No )
5 ssltss2 27129 . . . . . . . . 9 (𝐶 <<s 𝐷𝐷 No )
65ad2antlr 725 . . . . . . . 8 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐷 No )
76sselda 3944 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → 𝑑 No )
8 simplr 767 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷))
9 scutcut 27140 . . . . . . . . . . . 12 (𝐶 <<s 𝐷 → ((𝐶 |s 𝐷) ∈ No 𝐶 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐷))
109simp3d 1144 . . . . . . . . . . 11 (𝐶 <<s 𝐷 → {(𝐶 |s 𝐷)} <<s 𝐷)
1110ad2antlr 725 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → {(𝐶 |s 𝐷)} <<s 𝐷)
12 ssltsep 27130 . . . . . . . . . 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 7390 . . . . . . . . . 10 (𝐶 |s 𝐷) ∈ V
15 breq1 5108 . . . . . . . . . . 11 (𝑎 = (𝐶 |s 𝐷) → (𝑎 <s 𝑑 ↔ (𝐶 |s 𝐷) <s 𝑑))
1615ralbidv 3174 . . . . . . . . . 10 (𝑎 = (𝐶 |s 𝐷) → (∀𝑑𝐷 𝑎 <s 𝑑 ↔ ∀𝑑𝐷 (𝐶 |s 𝐷) <s 𝑑))
1714, 16ralsn 4642 . . . . . . . . 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 3234 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐶 |s 𝐷) <s 𝑑)
202, 4, 7, 8, 19slelttrd 27109 . . . . . 6 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐴 |s 𝐵) <s 𝑑)
2120ralrimiva 3143 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑)
22 ssltss1 27128 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐴 No )
2322adantr 481 . . . . . . . . 9 ((𝐴 <<s 𝐵𝐶 <<s 𝐷) → 𝐴 No )
2423adantr 481 . . . . . . . 8 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐴 No )
2524sselda 3944 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → 𝑎 No )
261ad3antrrr 728 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → (𝐴 |s 𝐵) ∈ No )
273ad3antlr 729 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → (𝐶 |s 𝐷) ∈ No )
28 scutcut 27140 . . . . . . . . . . . . 13 (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
2928simp2d 1143 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐴 <<s {(𝐴 |s 𝐵)})
3029adantr 481 . . . . . . . . . . 11 ((𝐴 <<s 𝐵𝐶 <<s 𝐷) → 𝐴 <<s {(𝐴 |s 𝐵)})
3130adantr 481 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐴 <<s {(𝐴 |s 𝐵)})
32 ssltsep 27130 . . . . . . . . . 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 3234 . . . . . . . 8 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → ∀𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑)
35 ovex 7390 . . . . . . . . 9 (𝐴 |s 𝐵) ∈ V
36 breq2 5109 . . . . . . . . 9 (𝑑 = (𝐴 |s 𝐵) → (𝑎 <s 𝑑𝑎 <s (𝐴 |s 𝐵)))
3735, 36ralsn 4642 . . . . . . . 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 767 . . . . . . 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 27108 . . . . . 6 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → 𝑎 <s (𝐶 |s 𝐷))
4140ralrimiva 3143 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))
4221, 41jca 512 . . . 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 27116 . . . . . . 7 ( bday ‘(𝐴 |s 𝐵)) ∈ On
4443onordi 6428 . . . . . 6 Ord ( bday ‘(𝐴 |s 𝐵))
45 ordn2lp 6337 . . . . . 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 725 . . . . . . 7 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) → (𝐶 |s 𝐷) ∈ No )
481adantr 481 . . . . . . . 8 ((𝐴 <<s 𝐵𝐶 <<s 𝐷) → (𝐴 |s 𝐵) ∈ No )
4948adantr 481 . . . . . . 7 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ∈ No )
50 sltnle 27101 . . . . . . 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 584 . . . . . 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 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)) → (𝐶 |s 𝐷) ∈ No )
53 ssltex1 27126 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐴 ∈ V)
5453ad3antrrr 728 . . . . . . . . . . 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 5388 . . . . . . . . . . 11 {(𝐶 |s 𝐷)} ∈ V
5654, 55jctir 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 ∈ V ∧ {(𝐶 |s 𝐷)} ∈ V))
5722ad3antrrr 728 . . . . . . . . . . 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 4769 . . . . . . . . . . 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 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)) → ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))
60 breq2 5109 . . . . . . . . . . . . . 14 (𝑑 = (𝐶 |s 𝐷) → (𝑎 <s 𝑑𝑎 <s (𝐶 |s 𝐷)))
6114, 60ralsn 4642 . . . . . . . . . . . . 13 (∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑𝑎 <s (𝐶 |s 𝐷))
6261ralbii 3096 . . . . . . . . . . . 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 1128 . . . . . . . . . 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 27125 . . . . . . . . . 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 583 . . . . . . . . 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 27127 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐵 ∈ V)
6867ad3antrrr 728 . . . . . . . . . . 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 520 . . . . . . . . . 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 27129 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐵 No )
7170ad3antrrr 728 . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . 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 3944 . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . 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 1144 . . . . . . . . . . . . . . . . . 18 (𝐴 <<s 𝐵 → {(𝐴 |s 𝐵)} <<s 𝐵)
7776ad3antrrr 728 . . . . . . . . . . . . . . . . 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 27130 . . . . . . . . . . . . . . . . 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 5108 . . . . . . . . . . . . . . . . . 18 (𝑎 = (𝐴 |s 𝐵) → (𝑎 <s 𝑏 ↔ (𝐴 |s 𝐵) <s 𝑏))
8180ralbidv 3174 . . . . . . . . . . . . . . . . 17 (𝑎 = (𝐴 |s 𝐵) → (∀𝑏𝐵 𝑎 <s 𝑏 ↔ ∀𝑏𝐵 (𝐴 |s 𝐵) <s 𝑏))
8235, 81ralsn 4642 . . . . . . . . . . . . . . . 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 3234 . . . . . . . . . . . . . 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 27107 . . . . . . . . . . . . 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 3143 . . . . . . . . . . . 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 5108 . . . . . . . . . . . . . 14 (𝑎 = (𝐶 |s 𝐷) → (𝑎 <s 𝑏 ↔ (𝐶 |s 𝐷) <s 𝑏))
8887ralbidv 3174 . . . . . . . . . . . . 13 (𝑎 = (𝐶 |s 𝐷) → (∀𝑏𝐵 𝑎 <s 𝑏 ↔ ∀𝑏𝐵 (𝐶 |s 𝐷) <s 𝑏))
8914, 88ralsn 4642 . . . . . . . . . . . 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 1128 . . . . . . . . . 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 27125 . . . . . . . . . 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 583 . . . . . . . . 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 27094 . . . . . . . . . . . . . 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 5108 . . . . . . . . . . . . . 14 ((𝐴 |s 𝐵) = (𝐶 |s 𝐷) → ((𝐴 |s 𝐵) <s (𝐴 |s 𝐵) ↔ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)))
9796notbid 317 . . . . . . . . . . . . 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 2958 . . . . . . . . . . 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 407 . . . . . . . . . 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 2999 . . . . . . . . 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 27157 . . . . . . . . 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 1375 . . . . . . . 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 728 . . . . . . . . 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 27126 . . . . . . . . . . . 12 (𝐶 <<s 𝐷𝐶 ∈ V)
106105ad3antlr 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)) → 𝐶 ∈ V)
107 snex 5388 . . . . . . . . . . 11 {(𝐴 |s 𝐵)} ∈ V
108106, 107jctir 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 ∈ V ∧ {(𝐴 |s 𝐵)} ∈ V))
109 ssltss1 27128 . . . . . . . . . . . 12 (𝐶 <<s 𝐷𝐶 No )
110109ad3antlr 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)) → 𝐶 No )
111104snssd 4769 . . . . . . . . . . 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 3944 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . 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 1143 . . . . . . . . . . . . . . . . . 18 (𝐶 <<s 𝐷𝐶 <<s {(𝐶 |s 𝐷)})
116115ad3antlr 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)) → 𝐶 <<s {(𝐶 |s 𝐷)})
117 ssltsep 27130 . . . . . . . . . . . . . . . . 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 3234 . . . . . . . . . . . . . . 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 5109 . . . . . . . . . . . . . . . 16 (𝑑 = (𝐶 |s 𝐷) → (𝑐 <s 𝑑𝑐 <s (𝐶 |s 𝐷)))
12114, 120ralsn 4642 . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . 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 27107 . . . . . . . . . . . . 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 5109 . . . . . . . . . . . . . 14 (𝑎 = (𝐴 |s 𝐵) → (𝑐 <s 𝑎𝑐 <s (𝐴 |s 𝐵)))
12635, 125ralsn 4642 . . . . . . . . . . . . 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 3143 . . . . . . . . . . 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 1128 . . . . . . . . . 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 27125 . . . . . . . . . 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 583 . . . . . . . . 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 27127 . . . . . . . . . . . 12 (𝐶 <<s 𝐷𝐷 ∈ V)
133132ad3antlr 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)) → 𝐷 ∈ V)
134133, 107jctil 520 . . . . . . . . . 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 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)) → 𝐷 No )
136 simplrl 775 . . . . . . . . . . . 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 5108 . . . . . . . . . . . . . 14 (𝑎 = (𝐴 |s 𝐵) → (𝑎 <s 𝑑 ↔ (𝐴 |s 𝐵) <s 𝑑))
138137ralbidv 3174 . . . . . . . . . . . . 13 (𝑎 = (𝐴 |s 𝐵) → (∀𝑑𝐷 𝑎 <s 𝑑 ↔ ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑))
13935, 138ralsn 4642 . . . . . . . . . . . 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 1128 . . . . . . . . . 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 27125 . . . . . . . . . 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 583 . . . . . . . . 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 27157 . . . . . . . . 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 1375 . . . . . . . 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 512 . . . . . . 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 413 . . . . . 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 259 . . . . 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 799 . . 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 5110 . . . 4 ((𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷)) → (𝑋 ≤s 𝑌 ↔ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)))
152 breq1 5108 . . . . . 6 (𝑋 = (𝐴 |s 𝐵) → (𝑋 <s 𝑑 ↔ (𝐴 |s 𝐵) <s 𝑑))
153152ralbidv 3174 . . . . 5 (𝑋 = (𝐴 |s 𝐵) → (∀𝑑𝐷 𝑋 <s 𝑑 ↔ ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑))
154 breq2 5109 . . . . . 6 (𝑌 = (𝐶 |s 𝐷) → (𝑎 <s 𝑌𝑎 <s (𝐶 |s 𝐷)))
155154ralbidv 3174 . . . . 5 (𝑌 = (𝐶 |s 𝐷) → (∀𝑎𝐴 𝑎 <s 𝑌 ↔ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷)))
156153, 155bi2anan9 637 . . . 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 345 . . 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, 157syl5ibr 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 408 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 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  Vcvv 3445  wss 3910  {csn 4586   class class class wbr 5105  Ord word 6316  cfv 6496  (class class class)co 7357   No csur 26988   <s cslt 26989   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:  sltrec  27159
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