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Theorem cofcutr 27243
Description: If 𝑋 is the cut of 𝐴 and 𝐵, then 𝐴 is cofinal with ( L ‘𝑋) and 𝐵 is coinitial with ( R ‘𝑋). Theorem 2.9 of [Gonshor] p. 12. (Contributed by Scott Fenton, 25-Sep-2024.)
Assertion
Ref Expression
cofcutr ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ ( L ‘𝑋)∃𝑦𝐴 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ ( R ‘𝑋)∃𝑤𝐵 𝑤 ≤s 𝑧))
Distinct variable groups:   𝑤,𝐴,𝑧   𝑥,𝐴,𝑦   𝑤,𝐵,𝑧   𝑥,𝐵,𝑦   𝑤,𝑋,𝑧   𝑥,𝑋,𝑦

Proof of Theorem cofcutr
Dummy variables 𝑎 𝑏 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bdayelon 27116 . . . . . . . . . 10 ( bday ‘(𝐴 |s 𝐵)) ∈ On
21onssneli 6433 . . . . . . . . 9 (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑥) → ¬ ( bday 𝑥) ∈ ( bday ‘(𝐴 |s 𝐵)))
3 leftssold 27208 . . . . . . . . . . . . 13 ( L ‘𝑋) ⊆ ( O ‘( bday 𝑋))
43a1i 11 . . . . . . . . . . . 12 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ( L ‘𝑋) ⊆ ( O ‘( bday 𝑋)))
54sselda 3944 . . . . . . . . . . 11 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → 𝑥 ∈ ( O ‘( bday 𝑋)))
6 bdayelon 27116 . . . . . . . . . . . 12 ( bday 𝑋) ∈ On
7 leftssno 27210 . . . . . . . . . . . . . 14 ( L ‘𝑋) ⊆ No
87a1i 11 . . . . . . . . . . . . 13 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ( L ‘𝑋) ⊆ No )
98sselda 3944 . . . . . . . . . . . 12 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → 𝑥 No )
10 oldbday 27230 . . . . . . . . . . . 12 ((( bday 𝑋) ∈ On ∧ 𝑥 No ) → (𝑥 ∈ ( O ‘( bday 𝑋)) ↔ ( bday 𝑥) ∈ ( bday 𝑋)))
116, 9, 10sylancr 587 . . . . . . . . . . 11 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → (𝑥 ∈ ( O ‘( bday 𝑋)) ↔ ( bday 𝑥) ∈ ( bday 𝑋)))
125, 11mpbid 231 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ( bday 𝑥) ∈ ( bday 𝑋))
13 simplr 767 . . . . . . . . . . 11 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → 𝑋 = (𝐴 |s 𝐵))
1413fveq2d 6846 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ( bday 𝑋) = ( bday ‘(𝐴 |s 𝐵)))
1512, 14eleqtrd 2840 . . . . . . . . 9 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ( bday 𝑥) ∈ ( bday ‘(𝐴 |s 𝐵)))
162, 15nsyl3 138 . . . . . . . 8 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ¬ ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑥))
17 scutbday 27143 . . . . . . . . . 10 (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}))
1817ad3antrrr 728 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝐴 <<s {𝑥}) → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}))
19 bdayfn 27113 . . . . . . . . . . 11 bday Fn No
20 ssrab2 4037 . . . . . . . . . . 11 {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)} ⊆ No
21 sneq 4596 . . . . . . . . . . . . . 14 (𝑡 = 𝑥 → {𝑡} = {𝑥})
2221breq2d 5117 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → (𝐴 <<s {𝑡} ↔ 𝐴 <<s {𝑥}))
2321breq1d 5115 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → ({𝑡} <<s 𝐵 ↔ {𝑥} <<s 𝐵))
2422, 23anbi12d 631 . . . . . . . . . . . 12 (𝑡 = 𝑥 → ((𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵) ↔ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)))
259adantr 481 . . . . . . . . . . . 12 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝐴 <<s {𝑥}) → 𝑥 No )
26 vsnex 5386 . . . . . . . . . . . . . . 15 {𝑥} ∈ V
2726a1i 11 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → {𝑥} ∈ V)
28 ssltex2 27127 . . . . . . . . . . . . . . 15 (𝐴 <<s 𝐵𝐵 ∈ V)
2928ad2antrr 724 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → 𝐵 ∈ V)
309snssd 4769 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → {𝑥} ⊆ No )
31 ssltss2 27129 . . . . . . . . . . . . . . 15 (𝐴 <<s 𝐵𝐵 No )
3231ad2antrr 724 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → 𝐵 No )
339adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → 𝑥 No )
34 simpr 485 . . . . . . . . . . . . . . . . . . 19 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → 𝑋 = (𝐴 |s 𝐵))
35 simpl 483 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → 𝐴 <<s 𝐵)
3635scutcld 27142 . . . . . . . . . . . . . . . . . . 19 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (𝐴 |s 𝐵) ∈ No )
3734, 36eqeltrd 2838 . . . . . . . . . . . . . . . . . 18 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → 𝑋 No )
3837ad2antrr 724 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → 𝑋 No )
3932sselda 3944 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → 𝑏 No )
40 leftval 27193 . . . . . . . . . . . . . . . . . . . . . 22 ( L ‘𝑋) = {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋}
4140a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ( L ‘𝑋) = {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋})
4241eleq2d 2823 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (𝑥 ∈ ( L ‘𝑋) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋}))
43 rabid 3427 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋} ↔ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋))
4442, 43bitrdi 286 . . . . . . . . . . . . . . . . . . 19 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (𝑥 ∈ ( L ‘𝑋) ↔ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋)))
4544simplbda 500 . . . . . . . . . . . . . . . . . 18 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → 𝑥 <s 𝑋)
4645adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → 𝑥 <s 𝑋)
47 simpllr 774 . . . . . . . . . . . . . . . . . 18 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → 𝑋 = (𝐴 |s 𝐵))
48 scutcut 27140 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
4948ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
5049simp3d 1144 . . . . . . . . . . . . . . . . . . 19 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → {(𝐴 |s 𝐵)} <<s 𝐵)
51 ovex 7390 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 |s 𝐵) ∈ V
5251snid 4622 . . . . . . . . . . . . . . . . . . . 20 (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)}
53 ssltsepc 27132 . . . . . . . . . . . . . . . . . . . 20 (({(𝐴 |s 𝐵)} <<s 𝐵 ∧ (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)} ∧ 𝑏𝐵) → (𝐴 |s 𝐵) <s 𝑏)
5452, 53mp3an2 1449 . . . . . . . . . . . . . . . . . . 19 (({(𝐴 |s 𝐵)} <<s 𝐵𝑏𝐵) → (𝐴 |s 𝐵) <s 𝑏)
5550, 54sylan 580 . . . . . . . . . . . . . . . . . 18 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → (𝐴 |s 𝐵) <s 𝑏)
5647, 55eqbrtrd 5127 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → 𝑋 <s 𝑏)
5733, 38, 39, 46, 56slttrd 27107 . . . . . . . . . . . . . . . 16 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → 𝑥 <s 𝑏)
58573adant2 1131 . . . . . . . . . . . . . . 15 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑎 ∈ {𝑥} ∧ 𝑏𝐵) → 𝑥 <s 𝑏)
59 velsn 4602 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ {𝑥} ↔ 𝑎 = 𝑥)
60 breq1 5108 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → (𝑎 <s 𝑏𝑥 <s 𝑏))
6159, 60sylbi 216 . . . . . . . . . . . . . . . 16 (𝑎 ∈ {𝑥} → (𝑎 <s 𝑏𝑥 <s 𝑏))
62613ad2ant2 1134 . . . . . . . . . . . . . . 15 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑎 ∈ {𝑥} ∧ 𝑏𝐵) → (𝑎 <s 𝑏𝑥 <s 𝑏))
6358, 62mpbird 256 . . . . . . . . . . . . . 14 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑎 ∈ {𝑥} ∧ 𝑏𝐵) → 𝑎 <s 𝑏)
6427, 29, 30, 32, 63ssltd 27131 . . . . . . . . . . . . 13 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → {𝑥} <<s 𝐵)
6564anim1ci 616 . . . . . . . . . . . 12 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝐴 <<s {𝑥}) → (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))
6624, 25, 65elrabd 3647 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝐴 <<s {𝑥}) → 𝑥 ∈ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)})
67 fnfvima 7183 . . . . . . . . . . 11 (( bday Fn No ∧ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)} ⊆ No 𝑥 ∈ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}) → ( bday 𝑥) ∈ ( bday “ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}))
6819, 20, 66, 67mp3an12i 1465 . . . . . . . . . 10 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝐴 <<s {𝑥}) → ( bday 𝑥) ∈ ( bday “ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}))
69 intss1 4924 . . . . . . . . . 10 (( bday 𝑥) ∈ ( bday “ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}) → ( bday “ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}) ⊆ ( bday 𝑥))
7068, 69syl 17 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝐴 <<s {𝑥}) → ( bday “ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}) ⊆ ( bday 𝑥))
7118, 70eqsstrd 3982 . . . . . . . 8 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝐴 <<s {𝑥}) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑥))
7216, 71mtand 814 . . . . . . 7 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ¬ 𝐴 <<s {𝑥})
73 ssltex1 27126 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐴 ∈ V)
7473ad3antrrr 728 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡) → 𝐴 ∈ V)
7574, 26jctir 521 . . . . . . . 8 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡) → (𝐴 ∈ V ∧ {𝑥} ∈ V))
76 ssltss1 27128 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐴 No )
7776ad3antrrr 728 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡) → 𝐴 No )
789adantr 481 . . . . . . . . . 10 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡) → 𝑥 No )
7978snssd 4769 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡) → {𝑥} ⊆ No )
80 simpr 485 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡) → ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡)
8177, 79, 803jca 1128 . . . . . . . 8 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡) → (𝐴 No ∧ {𝑥} ⊆ No ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡))
82 brsslt 27125 . . . . . . . 8 (𝐴 <<s {𝑥} ↔ ((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 No ∧ {𝑥} ⊆ No ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡)))
8375, 81, 82sylanbrc 583 . . . . . . 7 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡) → 𝐴 <<s {𝑥})
8472, 83mtand 814 . . . . . 6 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ¬ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡)
85 rexnal 3103 . . . . . . 7 (∃𝑡 ∈ {𝑥} ¬ ∀𝑦𝐴 𝑦 <s 𝑡 ↔ ¬ ∀𝑡 ∈ {𝑥}∀𝑦𝐴 𝑦 <s 𝑡)
86 ralcom 3272 . . . . . . 7 (∀𝑡 ∈ {𝑥}∀𝑦𝐴 𝑦 <s 𝑡 ↔ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡)
8785, 86xchbinx 333 . . . . . 6 (∃𝑡 ∈ {𝑥} ¬ ∀𝑦𝐴 𝑦 <s 𝑡 ↔ ¬ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡)
8884, 87sylibr 233 . . . . 5 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ∃𝑡 ∈ {𝑥} ¬ ∀𝑦𝐴 𝑦 <s 𝑡)
89 vex 3449 . . . . . 6 𝑥 ∈ V
90 breq2 5109 . . . . . . . 8 (𝑡 = 𝑥 → (𝑦 <s 𝑡𝑦 <s 𝑥))
9190ralbidv 3174 . . . . . . 7 (𝑡 = 𝑥 → (∀𝑦𝐴 𝑦 <s 𝑡 ↔ ∀𝑦𝐴 𝑦 <s 𝑥))
9291notbid 317 . . . . . 6 (𝑡 = 𝑥 → (¬ ∀𝑦𝐴 𝑦 <s 𝑡 ↔ ¬ ∀𝑦𝐴 𝑦 <s 𝑥))
9389, 92rexsn 4643 . . . . 5 (∃𝑡 ∈ {𝑥} ¬ ∀𝑦𝐴 𝑦 <s 𝑡 ↔ ¬ ∀𝑦𝐴 𝑦 <s 𝑥)
9488, 93sylib 217 . . . 4 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ¬ ∀𝑦𝐴 𝑦 <s 𝑥)
9576ad2antrr 724 . . . . . . . 8 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → 𝐴 No )
9695sselda 3944 . . . . . . 7 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑦𝐴) → 𝑦 No )
97 slenlt 27100 . . . . . . 7 ((𝑥 No 𝑦 No ) → (𝑥 ≤s 𝑦 ↔ ¬ 𝑦 <s 𝑥))
989, 96, 97syl2an2r 683 . . . . . 6 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑦𝐴) → (𝑥 ≤s 𝑦 ↔ ¬ 𝑦 <s 𝑥))
9998rexbidva 3173 . . . . 5 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → (∃𝑦𝐴 𝑥 ≤s 𝑦 ↔ ∃𝑦𝐴 ¬ 𝑦 <s 𝑥))
100 rexnal 3103 . . . . 5 (∃𝑦𝐴 ¬ 𝑦 <s 𝑥 ↔ ¬ ∀𝑦𝐴 𝑦 <s 𝑥)
10199, 100bitrdi 286 . . . 4 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → (∃𝑦𝐴 𝑥 ≤s 𝑦 ↔ ¬ ∀𝑦𝐴 𝑦 <s 𝑥))
10294, 101mpbird 256 . . 3 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ∃𝑦𝐴 𝑥 ≤s 𝑦)
103102ralrimiva 3143 . 2 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ∀𝑥 ∈ ( L ‘𝑋)∃𝑦𝐴 𝑥 ≤s 𝑦)
1041onssneli 6433 . . . . . . . . 9 (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑧) → ¬ ( bday 𝑧) ∈ ( bday ‘(𝐴 |s 𝐵)))
105 rightssold 27209 . . . . . . . . . . . . 13 ( R ‘𝑋) ⊆ ( O ‘( bday 𝑋))
106105a1i 11 . . . . . . . . . . . 12 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ( R ‘𝑋) ⊆ ( O ‘( bday 𝑋)))
107106sselda 3944 . . . . . . . . . . 11 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝑧 ∈ ( O ‘( bday 𝑋)))
108 rightssno 27211 . . . . . . . . . . . . . 14 ( R ‘𝑋) ⊆ No
109108a1i 11 . . . . . . . . . . . . 13 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ( R ‘𝑋) ⊆ No )
110109sselda 3944 . . . . . . . . . . . 12 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝑧 No )
111 oldbday 27230 . . . . . . . . . . . 12 ((( bday 𝑋) ∈ On ∧ 𝑧 No ) → (𝑧 ∈ ( O ‘( bday 𝑋)) ↔ ( bday 𝑧) ∈ ( bday 𝑋)))
1126, 110, 111sylancr 587 . . . . . . . . . . 11 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → (𝑧 ∈ ( O ‘( bday 𝑋)) ↔ ( bday 𝑧) ∈ ( bday 𝑋)))
113107, 112mpbid 231 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ( bday 𝑧) ∈ ( bday 𝑋))
114 simplr 767 . . . . . . . . . . 11 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝑋 = (𝐴 |s 𝐵))
115114fveq2d 6846 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ( bday 𝑋) = ( bday ‘(𝐴 |s 𝐵)))
116113, 115eleqtrd 2840 . . . . . . . . 9 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ( bday 𝑧) ∈ ( bday ‘(𝐴 |s 𝐵)))
117104, 116nsyl3 138 . . . . . . . 8 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ¬ ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑧))
11817ad3antrrr 728 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ {𝑧} <<s 𝐵) → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}))
119 sneq 4596 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → {𝑡} = {𝑧})
120119breq2d 5117 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (𝐴 <<s {𝑡} ↔ 𝐴 <<s {𝑧}))
121119breq1d 5115 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → ({𝑡} <<s 𝐵 ↔ {𝑧} <<s 𝐵))
122120, 121anbi12d 631 . . . . . . . . . . . 12 (𝑡 = 𝑧 → ((𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵) ↔ (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵)))
123110adantr 481 . . . . . . . . . . . 12 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ {𝑧} <<s 𝐵) → 𝑧 No )
12473ad2antrr 724 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝐴 ∈ V)
125 vsnex 5386 . . . . . . . . . . . . . . 15 {𝑧} ∈ V
126125a1i 11 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → {𝑧} ∈ V)
12776ad2antrr 724 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝐴 No )
128110snssd 4769 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → {𝑧} ⊆ No )
129127sselda 3944 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑎 No )
13037ad2antrr 724 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑋 No )
131110adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑧 No )
13248ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
133132simp2d 1143 . . . . . . . . . . . . . . . . . . 19 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝐴 <<s {(𝐴 |s 𝐵)})
134 ssltsepc 27132 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 <<s {(𝐴 |s 𝐵)} ∧ 𝑎𝐴 ∧ (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)}) → 𝑎 <s (𝐴 |s 𝐵))
13552, 134mp3an3 1450 . . . . . . . . . . . . . . . . . . 19 ((𝐴 <<s {(𝐴 |s 𝐵)} ∧ 𝑎𝐴) → 𝑎 <s (𝐴 |s 𝐵))
136133, 135sylan 580 . . . . . . . . . . . . . . . . . 18 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑎 <s (𝐴 |s 𝐵))
137 simpllr 774 . . . . . . . . . . . . . . . . . 18 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑋 = (𝐴 |s 𝐵))
138136, 137breqtrrd 5133 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑎 <s 𝑋)
139 rightval 27194 . . . . . . . . . . . . . . . . . . . . . 22 ( R ‘𝑋) = {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}
140139a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ( R ‘𝑋) = {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧})
141140eleq2d 2823 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (𝑧 ∈ ( R ‘𝑋) ↔ 𝑧 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}))
142 rabid 3427 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧} ↔ (𝑧 ∈ ( O ‘( bday 𝑋)) ∧ 𝑋 <s 𝑧))
143141, 142bitrdi 286 . . . . . . . . . . . . . . . . . . 19 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (𝑧 ∈ ( R ‘𝑋) ↔ (𝑧 ∈ ( O ‘( bday 𝑋)) ∧ 𝑋 <s 𝑧)))
144143simplbda 500 . . . . . . . . . . . . . . . . . 18 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝑋 <s 𝑧)
145144adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑋 <s 𝑧)
146129, 130, 131, 138, 145slttrd 27107 . . . . . . . . . . . . . . . 16 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑎 <s 𝑧)
1471463adant3 1132 . . . . . . . . . . . . . . 15 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴𝑏 ∈ {𝑧}) → 𝑎 <s 𝑧)
148 velsn 4602 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ {𝑧} ↔ 𝑏 = 𝑧)
149 breq2 5109 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑧 → (𝑎 <s 𝑏𝑎 <s 𝑧))
150148, 149sylbi 216 . . . . . . . . . . . . . . . 16 (𝑏 ∈ {𝑧} → (𝑎 <s 𝑏𝑎 <s 𝑧))
1511503ad2ant3 1135 . . . . . . . . . . . . . . 15 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴𝑏 ∈ {𝑧}) → (𝑎 <s 𝑏𝑎 <s 𝑧))
152147, 151mpbird 256 . . . . . . . . . . . . . 14 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴𝑏 ∈ {𝑧}) → 𝑎 <s 𝑏)
153124, 126, 127, 128, 152ssltd 27131 . . . . . . . . . . . . 13 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝐴 <<s {𝑧})
154153anim1i 615 . . . . . . . . . . . 12 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ {𝑧} <<s 𝐵) → (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵))
155122, 123, 154elrabd 3647 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ {𝑧} <<s 𝐵) → 𝑧 ∈ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)})
156 fnfvima 7183 . . . . . . . . . . 11 (( bday Fn No ∧ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)} ⊆ No 𝑧 ∈ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}) → ( bday 𝑧) ∈ ( bday “ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}))
15719, 20, 155, 156mp3an12i 1465 . . . . . . . . . 10 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ {𝑧} <<s 𝐵) → ( bday 𝑧) ∈ ( bday “ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}))
158 intss1 4924 . . . . . . . . . 10 (( bday 𝑧) ∈ ( bday “ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}) → ( bday “ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}) ⊆ ( bday 𝑧))
159157, 158syl 17 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ {𝑧} <<s 𝐵) → ( bday “ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}) ⊆ ( bday 𝑧))
160118, 159eqsstrd 3982 . . . . . . . 8 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ {𝑧} <<s 𝐵) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑧))
161117, 160mtand 814 . . . . . . 7 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ¬ {𝑧} <<s 𝐵)
16228ad3antrrr 728 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤) → 𝐵 ∈ V)
163162, 125jctil 520 . . . . . . . 8 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤) → ({𝑧} ∈ V ∧ 𝐵 ∈ V))
164128adantr 481 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤) → {𝑧} ⊆ No )
16531ad3antrrr 728 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤) → 𝐵 No )
166 simpr 485 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤) → ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤)
167164, 165, 1663jca 1128 . . . . . . . 8 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤) → ({𝑧} ⊆ No 𝐵 No ∧ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤))
168 brsslt 27125 . . . . . . . 8 ({𝑧} <<s 𝐵 ↔ (({𝑧} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑧} ⊆ No 𝐵 No ∧ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤)))
169163, 167, 168sylanbrc 583 . . . . . . 7 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤) → {𝑧} <<s 𝐵)
170161, 169mtand 814 . . . . . 6 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ¬ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤)
171 rexnal 3103 . . . . . 6 (∃𝑡 ∈ {𝑧} ¬ ∀𝑤𝐵 𝑡 <s 𝑤 ↔ ¬ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤)
172170, 171sylibr 233 . . . . 5 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ∃𝑡 ∈ {𝑧} ¬ ∀𝑤𝐵 𝑡 <s 𝑤)
173 vex 3449 . . . . . 6 𝑧 ∈ V
174 breq1 5108 . . . . . . . 8 (𝑡 = 𝑧 → (𝑡 <s 𝑤𝑧 <s 𝑤))
175174ralbidv 3174 . . . . . . 7 (𝑡 = 𝑧 → (∀𝑤𝐵 𝑡 <s 𝑤 ↔ ∀𝑤𝐵 𝑧 <s 𝑤))
176175notbid 317 . . . . . 6 (𝑡 = 𝑧 → (¬ ∀𝑤𝐵 𝑡 <s 𝑤 ↔ ¬ ∀𝑤𝐵 𝑧 <s 𝑤))
177173, 176rexsn 4643 . . . . 5 (∃𝑡 ∈ {𝑧} ¬ ∀𝑤𝐵 𝑡 <s 𝑤 ↔ ¬ ∀𝑤𝐵 𝑧 <s 𝑤)
178172, 177sylib 217 . . . 4 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ¬ ∀𝑤𝐵 𝑧 <s 𝑤)
17931ad2antrr 724 . . . . . . . 8 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝐵 No )
180179sselda 3944 . . . . . . 7 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑤𝐵) → 𝑤 No )
181110adantr 481 . . . . . . 7 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑤𝐵) → 𝑧 No )
182 slenlt 27100 . . . . . . 7 ((𝑤 No 𝑧 No ) → (𝑤 ≤s 𝑧 ↔ ¬ 𝑧 <s 𝑤))
183180, 181, 182syl2anc 584 . . . . . 6 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑤𝐵) → (𝑤 ≤s 𝑧 ↔ ¬ 𝑧 <s 𝑤))
184183rexbidva 3173 . . . . 5 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → (∃𝑤𝐵 𝑤 ≤s 𝑧 ↔ ∃𝑤𝐵 ¬ 𝑧 <s 𝑤))
185 rexnal 3103 . . . . 5 (∃𝑤𝐵 ¬ 𝑧 <s 𝑤 ↔ ¬ ∀𝑤𝐵 𝑧 <s 𝑤)
186184, 185bitrdi 286 . . . 4 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → (∃𝑤𝐵 𝑤 ≤s 𝑧 ↔ ¬ ∀𝑤𝐵 𝑧 <s 𝑤))
187178, 186mpbird 256 . . 3 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ∃𝑤𝐵 𝑤 ≤s 𝑧)
188187ralrimiva 3143 . 2 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ∀𝑧 ∈ ( R ‘𝑋)∃𝑤𝐵 𝑤 ≤s 𝑧)
189103, 188jca 512 1 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ ( L ‘𝑋)∃𝑦𝐴 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ ( R ‘𝑋)∃𝑤𝐵 𝑤 ≤s 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  wss 3910  {csn 4586   cint 4907   class class class wbr 5105  cima 5636  Oncon0 6317   Fn wfn 6491  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   O cold 27173   L cleft 27175   R cright 27176
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-pow 5320  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-pred 6253  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-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-1o 8412  df-2o 8413  df-no 26991  df-slt 26992  df-bday 26993  df-sle 27093  df-sslt 27121  df-scut 27123  df-made 27177  df-old 27178  df-left 27180  df-right 27181
This theorem is referenced by:  cofcutr1d  27244  cofcutr2d  27245
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