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Theorem slerec 32249
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 32238 . . . . . . . . 9 (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
21simp1d 1165 . . . . . . . 8 (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
32ad3antrrr 712 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐴 |s 𝐵) ∈ No )
4 scutcut 32238 . . . . . . . . 9 (𝐶 <<s 𝐷 → ((𝐶 |s 𝐷) ∈ No 𝐶 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐷))
54simp1d 1165 . . . . . . . 8 (𝐶 <<s 𝐷 → (𝐶 |s 𝐷) ∈ No )
65ad3antlr 713 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐶 |s 𝐷) ∈ No )
7 ssltss2 32230 . . . . . . . . 9 (𝐶 <<s 𝐷𝐷 No )
87ad2antlr 709 . . . . . . . 8 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐷 No )
98sselda 3805 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → 𝑑 No )
10 simplr 776 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷))
114simp3d 1167 . . . . . . . . . . 11 (𝐶 <<s 𝐷 → {(𝐶 |s 𝐷)} <<s 𝐷)
1211ad2antlr 709 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → {(𝐶 |s 𝐷)} <<s 𝐷)
13 ssltsep 32231 . . . . . . . . . 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 6909 . . . . . . . . . 10 (𝐶 |s 𝐷) ∈ V
16 breq1 4854 . . . . . . . . . . 11 (𝑎 = (𝐶 |s 𝐷) → (𝑎 <s 𝑑 ↔ (𝐶 |s 𝐷) <s 𝑑))
1716ralbidv 3181 . . . . . . . . . 10 (𝑎 = (𝐶 |s 𝐷) → (∀𝑑𝐷 𝑎 <s 𝑑 ↔ ∀𝑑𝐷 (𝐶 |s 𝐷) <s 𝑑))
1815, 17ralsn 4422 . . . . . . . . 9 (∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑑𝐷 𝑎 <s 𝑑 ↔ ∀𝑑𝐷 (𝐶 |s 𝐷) <s 𝑑)
1914, 18sylib 209 . . . . . . . 8 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑑𝐷 (𝐶 |s 𝐷) <s 𝑑)
2019r19.21bi 3127 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐶 |s 𝐷) <s 𝑑)
213, 6, 9, 10, 20slelttrd 32212 . . . . . 6 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑𝐷) → (𝐴 |s 𝐵) <s 𝑑)
2221ralrimiva 3161 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑)
23 ssltss1 32229 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐴 No )
2423adantr 468 . . . . . . . . 9 ((𝐴 <<s 𝐵𝐶 <<s 𝐷) → 𝐴 No )
2524adantr 468 . . . . . . . 8 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐴 No )
2625sselda 3805 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → 𝑎 No )
272ad3antrrr 712 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → (𝐴 |s 𝐵) ∈ No )
285ad3antlr 713 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → (𝐶 |s 𝐷) ∈ No )
291simp2d 1166 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐴 <<s {(𝐴 |s 𝐵)})
3029adantr 468 . . . . . . . . . . 11 ((𝐴 <<s 𝐵𝐶 <<s 𝐷) → 𝐴 <<s {(𝐴 |s 𝐵)})
3130adantr 468 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐴 <<s {(𝐴 |s 𝐵)})
32 ssltsep 32231 . . . . . . . . . 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 3127 . . . . . . . 8 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → ∀𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑)
35 ovex 6909 . . . . . . . . 9 (𝐴 |s 𝐵) ∈ V
36 breq2 4855 . . . . . . . . 9 (𝑑 = (𝐴 |s 𝐵) → (𝑎 <s 𝑑𝑎 <s (𝐴 |s 𝐵)))
3735, 36ralsn 4422 . . . . . . . 8 (∀𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑𝑎 <s (𝐴 |s 𝐵))
3834, 37sylib 209 . . . . . . 7 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → 𝑎 <s (𝐴 |s 𝐵))
39 simplr 776 . . . . . . 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 32211 . . . . . 6 ((((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎𝐴) → 𝑎 <s (𝐶 |s 𝐷))
4140ralrimiva 3161 . . . . 5 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))
4222, 41jca 503 . . . 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 32218 . . . . . . 7 ( bday ‘(𝐴 |s 𝐵)) ∈ On
4443onordi 6048 . . . . . 6 Ord ( bday ‘(𝐴 |s 𝐵))
45 ordn2lp 5963 . . . . . 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 709 . . . . . . 7 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) → (𝐶 |s 𝐷) ∈ No )
482adantr 468 . . . . . . . 8 ((𝐴 <<s 𝐵𝐶 <<s 𝐷) → (𝐴 |s 𝐵) ∈ No )
4948adantr 468 . . . . . . 7 (((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ∈ No )
50 sltnle 32204 . . . . . . 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 575 . . . . . 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 713 . . . . . . . . 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 32227 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐴 ∈ V)
5453ad3antrrr 712 . . . . . . . . . . 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 5105 . . . . . . . . . . 11 {(𝐶 |s 𝐷)} ∈ V
5654, 55jctir 512 . . . . . . . . . 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 712 . . . . . . . . . . 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 4537 . . . . . . . . . . 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 787 . . . . . . . . . . . 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 4855 . . . . . . . . . . . . . 14 (𝑑 = (𝐶 |s 𝐷) → (𝑎 <s 𝑑𝑎 <s (𝐶 |s 𝐷)))
6115, 60ralsn 4422 . . . . . . . . . . . . 13 (∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑𝑎 <s (𝐶 |s 𝐷))
6261ralbii 3175 . . . . . . . . . . . 12 (∀𝑎𝐴𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑 ↔ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷))
6359, 62sylibr 225 . . . . . . . . . . 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 1151 . . . . . . . . . 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 32226 . . . . . . . . . 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 574 . . . . . . . . 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 32228 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐵 ∈ V)
6867ad3antrrr 712 . . . . . . . . . . 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 511 . . . . . . . . . 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 32230 . . . . . . . . . . . 12 (𝐴 <<s 𝐵𝐵 No )
7170ad3antrrr 712 . . . . . . . . . . 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 468 . . . . . . . . . . . . . 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 712 . . . . . . . . . . . . . 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 3805 . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . 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 1167 . . . . . . . . . . . . . . . . . 18 (𝐴 <<s 𝐵 → {(𝐴 |s 𝐵)} <<s 𝐵)
7776ad3antrrr 712 . . . . . . . . . . . . . . . . 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 32231 . . . . . . . . . . . . . . . . 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 4854 . . . . . . . . . . . . . . . . . 18 (𝑎 = (𝐴 |s 𝐵) → (𝑎 <s 𝑏 ↔ (𝐴 |s 𝐵) <s 𝑏))
8180ralbidv 3181 . . . . . . . . . . . . . . . . 17 (𝑎 = (𝐴 |s 𝐵) → (∀𝑏𝐵 𝑎 <s 𝑏 ↔ ∀𝑏𝐵 (𝐴 |s 𝐵) <s 𝑏))
8235, 81ralsn 4422 . . . . . . . . . . . . . . . 16 (∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑏𝐵 𝑎 <s 𝑏 ↔ ∀𝑏𝐵 (𝐴 |s 𝐵) <s 𝑏)
8379, 82sylib 209 . . . . . . . . . . . . . . 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 3127 . . . . . . . . . . . . . 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 32210 . . . . . . . . . . . . 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 3161 . . . . . . . . . . . 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 4854 . . . . . . . . . . . . . 14 (𝑎 = (𝐶 |s 𝐷) → (𝑎 <s 𝑏 ↔ (𝐶 |s 𝐷) <s 𝑏))
8887ralbidv 3181 . . . . . . . . . . . . 13 (𝑎 = (𝐶 |s 𝐷) → (∀𝑏𝐵 𝑎 <s 𝑏 ↔ ∀𝑏𝐵 (𝐶 |s 𝐷) <s 𝑏))
8915, 88ralsn 4422 . . . . . . . . . . . 12 (∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑏𝐵 𝑎 <s 𝑏 ↔ ∀𝑏𝐵 (𝐶 |s 𝐷) <s 𝑏)
9086, 89sylibr 225 . . . . . . . . . . 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 1151 . . . . . . . . . 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 32226 . . . . . . . . . 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 574 . . . . . . . . 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 32197 . . . . . . . . . . . . . 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 4854 . . . . . . . . . . . . . 14 ((𝐴 |s 𝐵) = (𝐶 |s 𝐷) → ((𝐴 |s 𝐵) <s (𝐴 |s 𝐵) ↔ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)))
9796notbid 309 . . . . . . . . . . . . 13 ((𝐴 |s 𝐵) = (𝐶 |s 𝐷) → (¬ (𝐴 |s 𝐵) <s (𝐴 |s 𝐵) ↔ ¬ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)))
9895, 97syl5ibcom 236 . . . . . . . . . . . 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 3000 . . . . . . . . . . 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 395 . . . . . . . . . 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 3040 . . . . . . . . 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 32248 . . . . . . . . 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 1487 . . . . . . . 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 712 . . . . . . . . 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 32227 . . . . . . . . . . . 12 (𝐶 <<s 𝐷𝐶 ∈ V)
106105ad3antlr 713 . . . . . . . . . . 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 5105 . . . . . . . . . . 11 {(𝐴 |s 𝐵)} ∈ V
108106, 107jctir 512 . . . . . . . . . 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 32229 . . . . . . . . . . . 12 (𝐶 <<s 𝐷𝐶 No )
110109ad3antlr 713 . . . . . . . . . . 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 4537 . . . . . . . . . . 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 3805 . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . 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 712 . . . . . . . . . . . . . 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 1166 . . . . . . . . . . . . . . . . . 18 (𝐶 <<s 𝐷𝐶 <<s {(𝐶 |s 𝐷)})
116115ad3antlr 713 . . . . . . . . . . . . . . . . 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 32231 . . . . . . . . . . . . . . . . 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 3127 . . . . . . . . . . . . . . 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 4855 . . . . . . . . . . . . . . . 16 (𝑑 = (𝐶 |s 𝐷) → (𝑐 <s 𝑑𝑐 <s (𝐶 |s 𝐷)))
12115, 120ralsn 4422 . . . . . . . . . . . . . . 15 (∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑐 <s 𝑑𝑐 <s (𝐶 |s 𝐷))
122119, 121sylib 209 . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . 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 32210 . . . . . . . . . . . . 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 4855 . . . . . . . . . . . . . 14 (𝑎 = (𝐴 |s 𝐵) → (𝑐 <s 𝑎𝑐 <s (𝐴 |s 𝐵)))
12635, 125ralsn 4422 . . . . . . . . . . . . 13 (∀𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎𝑐 <s (𝐴 |s 𝐵))
127124, 126sylibr 225 . . . . . . . . . . . 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 3161 . . . . . . . . . . 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 1151 . . . . . . . . . 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 32226 . . . . . . . . . 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 574 . . . . . . . . 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 32228 . . . . . . . . . . . 12 (𝐶 <<s 𝐷𝐷 ∈ V)
133132ad3antlr 713 . . . . . . . . . . 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 511 . . . . . . . . . 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 713 . . . . . . . . . . 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 786 . . . . . . . . . . . 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 4854 . . . . . . . . . . . . . 14 (𝑎 = (𝐴 |s 𝐵) → (𝑎 <s 𝑑 ↔ (𝐴 |s 𝐵) <s 𝑑))
138137ralbidv 3181 . . . . . . . . . . . . 13 (𝑎 = (𝐴 |s 𝐵) → (∀𝑑𝐷 𝑎 <s 𝑑 ↔ ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑))
13935, 138ralsn 4422 . . . . . . . . . . . 12 (∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑑𝐷 𝑎 <s 𝑑 ↔ ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑)
140136, 139sylibr 225 . . . . . . . . . . 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 1151 . . . . . . . . . 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 32226 . . . . . . . . . 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 574 . . . . . . . . 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 32248 . . . . . . . . 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 1487 . . . . . . . 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 503 . . . . . . 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 399 . . . . . 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 251 . . . . 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 143 . . . 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 826 . . 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 4856 . . . 4 ((𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷)) → (𝑋 ≤s 𝑌 ↔ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)))
152 breq1 4854 . . . . . 6 (𝑋 = (𝐴 |s 𝐵) → (𝑋 <s 𝑑 ↔ (𝐴 |s 𝐵) <s 𝑑))
153152ralbidv 3181 . . . . 5 (𝑋 = (𝐴 |s 𝐵) → (∀𝑑𝐷 𝑋 <s 𝑑 ↔ ∀𝑑𝐷 (𝐴 |s 𝐵) <s 𝑑))
154 breq2 4855 . . . . . 6 (𝑌 = (𝐶 |s 𝐷) → (𝑎 <s 𝑌𝑎 <s (𝐶 |s 𝐷)))
155154ralbidv 3181 . . . . 5 (𝑌 = (𝐶 |s 𝐷) → (∀𝑎𝐴 𝑎 <s 𝑌 ↔ ∀𝑎𝐴 𝑎 <s (𝐶 |s 𝐷)))
156153, 155bi2anan9 622 . . . 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 336 . . 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 237 . 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 396 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  wne 2985  wral 3103  Vcvv 3398  wss 3776  {csn 4377   class class class wbr 4851  Ord word 5942  cfv 6104  (class class class)co 6877   No csur 32119   <s cslt 32120   bday cbday 32121   ≤s csle 32195   <<s csslt 32222   |s cscut 32224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-ord 5946  df-on 5947  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-1o 7799  df-2o 7800  df-no 32122  df-slt 32123  df-bday 32124  df-sle 32196  df-sslt 32223  df-scut 32225
This theorem is referenced by:  sltrec  32250
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