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Theorem slerec 32250
 Description: A comparison law for surreals considered as cuts of sets of surreals. In Conway's treatment, this is the definition of less than or equal. (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 scutcut 32239 . . . . . . . . 9 (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
21simp1d 1137 . . . . . . . 8 (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
32ad3antrrr 768 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐴 |s 𝐵) ∈ No )
4 scutcut 32239 . . . . . . . . 9 (𝐶 <<s 𝐷 → ((𝐶 |s 𝐷) ∈ No 𝐶 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐷))
54simp1d 1137 . . . . . . . 8 (𝐶 <<s 𝐷 → (𝐶 |s 𝐷) ∈ No )
65ad3antlr 769 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐶 |s 𝐷) ∈ No )
7 ssltss2 32231 . . . . . . . . 9 (𝐶 <<s 𝐷𝐷 No )
87ad2antlr 765 . . . . . . . 8 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐷 No )
98sselda 3744 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → 𝑑 No )
10 simplr 809 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷))
114simp3d 1139 . . . . . . . . . . 11 (𝐶 <<s 𝐷 → {(𝐶 |s 𝐷)} <<s 𝐷)
1211ad2antlr 765 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → {(𝐶 |s 𝐷)} <<s 𝐷)
13 ssltsep 32232 . . . . . . . . . 10 ({(𝐶 |s 𝐷)} <<s 𝐷 → ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑑𝐷 𝑎 <s 𝑑)
1412, 13syl 17 . . . . . . . . 9 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑑𝐷 𝑎 <s 𝑑)
15 ovex 6842 . . . . . . . . . 10 (𝐶 |s 𝐷) ∈ V
16 breq1 4807 . . . . . . . . . . 11 (𝑎 = (𝐶 |s 𝐷) → (𝑎 <s 𝑑 ↔ (𝐶 |s 𝐷) <s 𝑑))
1716ralbidv 3124 . . . . . . . . . 10 (𝑎 = (𝐶 |s 𝐷) → (∀𝑑𝐷 𝑎 <s 𝑑 ↔ ∀𝑑𝐷 (𝐶 |s 𝐷) <s 𝑑))
1815, 17ralsn 4366 . . . . . . . . 9 (∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑑𝐷 𝑎 <s 𝑑 ↔ ∀𝑑𝐷 (𝐶 |s 𝐷) <s 𝑑)
1914, 18sylib 208 . . . . . . . 8 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑑𝐷 (𝐶 |s 𝐷) <s 𝑑)
2019r19.21bi 3070 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐶 |s 𝐷) <s 𝑑)
213, 6, 9, 10, 20slelttrd 32213 . . . . . 6 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐴 |s 𝐵) <s 𝑑)
2221ralrimiva 3104 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑)
23 ssltss1 32230 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐴 No )
2423adantr 472 . . . . . . . . 9 ((𝐴 <<s 𝐵𝐶 <<s 𝐷) → 𝐴 No )
2524adantr 472 . . . . . . . 8 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐴 No )
2625sselda 3744 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → 𝑎 No )
272ad3antrrr 768 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → (𝐴 |s 𝐵) ∈ No )
285ad3antlr 769 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → (𝐶 |s 𝐷) ∈ No )
291simp2d 1138 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐴 <<s {(𝐴 |s 𝐵)})
3029adantr 472 . . . . . . . . . . 11 ((𝐴 <<s 𝐵𝐶 <<s 𝐷) → 𝐴 <<s {(𝐴 |s 𝐵)})
3130adantr 472 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐴 <<s {(𝐴 |s 𝐵)})
32 ssltsep 32232 . . . . . . . . . 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 3070 . . . . . . . 8 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → ∀𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑)
35 ovex 6842 . . . . . . . . 9 (𝐴 |s 𝐵) ∈ V
36 breq2 4808 . . . . . . . . 9 (𝑑 = (𝐴 |s 𝐵) → (𝑎 <s 𝑑𝑎 <s (𝐴 |s 𝐵)))
3735, 36ralsn 4366 . . . . . . . 8 (∀𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑𝑎 <s (𝐴 |s 𝐵))
3834, 37sylib 208 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → 𝑎 <s (𝐴 |s 𝐵))
39 simplr 809 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷))
4026, 27, 28, 38, 39sltletrd 32212 . . . . . 6 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → 𝑎 <s (𝐶 |s 𝐷))
4140ralrimiva 3104 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))
4222, 41jca 555 . . . 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 32219 . . . . . . 7 ( bday ‘(𝐴 |s 𝐵)) ∈ On
4443onordi 5993 . . . . . 6 Ord ( bday ‘(𝐴 |s 𝐵))
45 ordn2lp 5904 . . . . . 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)))
475ad2antlr 765 . . . . . . 7 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) → (𝐶 |s 𝐷) ∈ No )
482adantr 472 . . . . . . . 8 ((𝐴 <<s 𝐵𝐶 <<s 𝐷) → (𝐴 |s 𝐵) ∈ No )
4948adantr 472 . . . . . . 7 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ∈ No )
50 sltnle 32205 . . . . . . 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 696 . . . . . 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)))
525ad3antlr 769 . . . . . . . . 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 32228 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐴 ∈ V)
5453ad3antrrr 768 . . . . . . . . . . 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 5057 . . . . . . . . . . 11 {(𝐶 |s 𝐷)} ∈ V
5654, 55jctir 562 . . . . . . . . . 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))
5723ad3antrrr 768 . . . . . . . . . . 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 4485 . . . . . . . . . . 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 820 . . . . . . . . . . . 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 4808 . . . . . . . . . . . . . 14 (𝑑 = (𝐶 |s 𝐷) → (𝑎 <s 𝑑𝑎 <s (𝐶 |s 𝐷)))
6115, 60ralsn 4366 . . . . . . . . . . . . 13 (∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑𝑎 <s (𝐶 |s 𝐷))
6261ralbii 3118 . . . . . . . . . . . 12 (∀𝑎𝐴𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑 ↔ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))
6359, 62sylibr 224 . . . . . . . . . . 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 1123 . . . . . . . . . 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 32227 . . . . . . . . . 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 701 . . . . . . . . 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 32229 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐵 ∈ V)
6867ad3antrrr 768 . . . . . . . . . . 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 561 . . . . . . . . . 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 32231 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐵 No )
7170ad3antrrr 768 . . . . . . . . . . 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 472 . . . . . . . . . . . . . 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 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) → (𝐴 |s 𝐵) ∈ No )
7471sselda 3744 . . . . . . . . . . . . . 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 809 . . . . . . . . . . . . . 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))
761simp3d 1139 . . . . . . . . . . . . . . . . . 18 (𝐴 <<s 𝐵 → {(𝐴 |s 𝐵)} <<s 𝐵)
7776ad3antrrr 768 . . . . . . . . . . . . . . . . 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 32232 . . . . . . . . . . . . . . . . 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 4807 . . . . . . . . . . . . . . . . . 18 (𝑎 = (𝐴 |s 𝐵) → (𝑎 <s 𝑏 ↔ (𝐴 |s 𝐵) <s 𝑏))
8180ralbidv 3124 . . . . . . . . . . . . . . . . 17 (𝑎 = (𝐴 |s 𝐵) → (∀𝑏𝐵 𝑎 <s 𝑏 ↔ ∀𝑏𝐵 (𝐴 |s 𝐵) <s 𝑏))
8235, 81ralsn 4366 . . . . . . . . . . . . . . . 16 (∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑏𝐵 𝑎 <s 𝑏 ↔ ∀𝑏𝐵 (𝐴 |s 𝐵) <s 𝑏)
8379, 82sylib 208 . . . . . . . . . . . . . . 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 3070 . . . . . . . . . . . . . 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 32211 . . . . . . . . . . . . 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 3104 . . . . . . . . . . . 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 4807 . . . . . . . . . . . . . 14 (𝑎 = (𝐶 |s 𝐷) → (𝑎 <s 𝑏 ↔ (𝐶 |s 𝐷) <s 𝑏))
8887ralbidv 3124 . . . . . . . . . . . . 13 (𝑎 = (𝐶 |s 𝐷) → (∀𝑏𝐵 𝑎 <s 𝑏 ↔ ∀𝑏𝐵 (𝐶 |s 𝐷) <s 𝑏))
8915, 88ralsn 4366 . . . . . . . . . . . 12 (∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑏𝐵 𝑎 <s 𝑏 ↔ ∀𝑏𝐵 (𝐶 |s 𝐷) <s 𝑏)
9086, 89sylibr 224 . . . . . . . . . . 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 1123 . . . . . . . . . 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 32227 . . . . . . . . . 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 701 . . . . . . . . 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 32198 . . . . . . . . . . . . . 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 4807 . . . . . . . . . . . . . 14 ((𝐴 |s 𝐵) = (𝐶 |s 𝐷) → ((𝐴 |s 𝐵) <s (𝐴 |s 𝐵) ↔ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)))
9796notbid 307 . . . . . . . . . . . . 13 ((𝐴 |s 𝐵) = (𝐶 |s 𝐷) → (¬ (𝐴 |s 𝐵) <s (𝐴 |s 𝐵) ↔ ¬ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)))
9895, 97syl5ibcom 235 . . . . . . . . . . . 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 2947 . . . . . . . . . . 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 444 . . . . . . . . . 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 2987 . . . . . . . . 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 32249 . . . . . . . . 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 1482 . . . . . . . 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)))
1042ad3antrrr 768 . . . . . . . . 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 32228 . . . . . . . . . . . 12 (𝐶 <<s 𝐷𝐶 ∈ V)
106105ad3antlr 769 . . . . . . . . . . 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 5057 . . . . . . . . . . 11 {(𝐴 |s 𝐵)} ∈ V
108106, 107jctir 562 . . . . . . . . . 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 32230 . . . . . . . . . . . 12 (𝐶 <<s 𝐷𝐶 No )
110109ad3antlr 769 . . . . . . . . . . 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 4485 . . . . . . . . . . 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 3744 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 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 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) → (𝐴 |s 𝐵) ∈ No )
1154simp2d 1138 . . . . . . . . . . . . . . . . . 18 (𝐶 <<s 𝐷𝐶 <<s {(𝐶 |s 𝐷)})
116115ad3antlr 769 . . . . . . . . . . . . . . . . 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 32232 . . . . . . . . . . . . . . . . 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 3070 . . . . . . . . . . . . . . 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 4808 . . . . . . . . . . . . . . . 16 (𝑑 = (𝐶 |s 𝐷) → (𝑐 <s 𝑑𝑐 <s (𝐶 |s 𝐷)))
12115, 120ralsn 4366 . . . . . . . . . . . . . . 15 (∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑐 <s 𝑑𝑐 <s (𝐶 |s 𝐷))
122119, 121sylib 208 . . . . . . . . . . . . . 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 809 . . . . . . . . . . . . . 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 32211 . . . . . . . . . . . . 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 4808 . . . . . . . . . . . . . 14 (𝑎 = (𝐴 |s 𝐵) → (𝑐 <s 𝑎𝑐 <s (𝐴 |s 𝐵)))
12635, 125ralsn 4366 . . . . . . . . . . . . 13 (∀𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎𝑐 <s (𝐴 |s 𝐵))
127124, 126sylibr 224 . . . . . . . . . . . 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 3104 . . . . . . . . . . 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 1123 . . . . . . . . . 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 32227 . . . . . . . . . 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 701 . . . . . . . . 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 32229 . . . . . . . . . . . 12 (𝐶 <<s 𝐷𝐷 ∈ V)
133132ad3antlr 769 . . . . . . . . . . 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 561 . . . . . . . . . 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))
1357ad3antlr 769 . . . . . . . . . . 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 819 . . . . . . . . . . . 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 4807 . . . . . . . . . . . . . 14 (𝑎 = (𝐴 |s 𝐵) → (𝑎 <s 𝑑 ↔ (𝐴 |s 𝐵) <s 𝑑))
138137ralbidv 3124 . . . . . . . . . . . . 13 (𝑎 = (𝐴 |s 𝐵) → (∀𝑑𝐷 𝑎 <s 𝑑 ↔ ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑))
13935, 138ralsn 4366 . . . . . . . . . . . 12 (∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑑𝐷 𝑎 <s 𝑑 ↔ ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑)
140136, 139sylibr 224 . . . . . . . . . . 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 1123 . . . . . . . . . 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 32227 . . . . . . . . . 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 701 . . . . . . . . 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 32249 . . . . . . . . 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 1482 . . . . . . . 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 555 . . . . . . 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 449 . . . . . 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 250 . . . . 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 141 . . . 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 913 . . 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 4809 . . . 4 ((𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷)) → (𝑋 ≤s 𝑌 ↔ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)))
152 breq1 4807 . . . . . 6 (𝑋 = (𝐴 |s 𝐵) → (𝑋 <s 𝑑 ↔ (𝐴 |s 𝐵) <s 𝑑))
153152ralbidv 3124 . . . . 5 (𝑋 = (𝐴 |s 𝐵) → (∀𝑑𝐷 𝑋 <s 𝑑 ↔ ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑))
154 breq2 4808 . . . . . 6 (𝑌 = (𝐶 |s 𝐷) → (𝑎 <s 𝑌𝑎 <s (𝐶 |s 𝐷)))
155154ralbidv 3124 . . . . 5 (𝑌 = (𝐶 |s 𝐷) → (∀𝑎𝐴 𝑎 <s 𝑌 ↔ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷)))
156153, 155bi2anan9 953 . . . 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 334 . . 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 236 . 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 445 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 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   ≠ wne 2932  ∀wral 3050  Vcvv 3340   ⊆ wss 3715  {csn 4321   class class class wbr 4804  Ord word 5883  ‘cfv 6049  (class class class)co 6814   No csur 32120
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