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Theorem cofcutr 33747
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 33626 . . . . . . . . . 10 ( bday ‘(𝐴 |s 𝐵)) ∈ On
21onssneli 6292 . . . . . . . . 9 (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑥) → ¬ ( bday 𝑥) ∈ ( bday ‘(𝐴 |s 𝐵)))
3 leftssold 33716 . . . . . . . . . . . . 13 ( L ‘𝑋) ⊆ ( O ‘( bday 𝑋))
43a1i 11 . . . . . . . . . . . 12 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ( L ‘𝑋) ⊆ ( O ‘( bday 𝑋)))
54sselda 3887 . . . . . . . . . . 11 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → 𝑥 ∈ ( O ‘( bday 𝑋)))
6 bdayelon 33626 . . . . . . . . . . . 12 ( bday 𝑋) ∈ On
7 leftssno 33718 . . . . . . . . . . . . . 14 ( L ‘𝑋) ⊆ No
87a1i 11 . . . . . . . . . . . . 13 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ( L ‘𝑋) ⊆ No )
98sselda 3887 . . . . . . . . . . . 12 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → 𝑥 No )
10 oldbday 33736 . . . . . . . . . . . 12 ((( bday 𝑋) ∈ On ∧ 𝑥 No ) → (𝑥 ∈ ( O ‘( bday 𝑋)) ↔ ( bday 𝑥) ∈ ( bday 𝑋)))
116, 9, 10sylancr 590 . . . . . . . . . . 11 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → (𝑥 ∈ ( O ‘( bday 𝑋)) ↔ ( bday 𝑥) ∈ ( bday 𝑋)))
125, 11mpbid 235 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ( bday 𝑥) ∈ ( bday 𝑋))
13 simplr 769 . . . . . . . . . . 11 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → 𝑋 = (𝐴 |s 𝐵))
1413fveq2d 6690 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ( bday 𝑋) = ( bday ‘(𝐴 |s 𝐵)))
1512, 14eleqtrd 2836 . . . . . . . . 9 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ( bday 𝑥) ∈ ( bday ‘(𝐴 |s 𝐵)))
162, 15nsyl3 140 . . . . . . . 8 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ¬ ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑥))
17 scutbday 33653 . . . . . . . . . 10 (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}))
1817ad3antrrr 730 . . . . . . . . 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 33623 . . . . . . . . . . 11 bday Fn No
20 ssrab2 3979 . . . . . . . . . . 11 {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)} ⊆ No
21 sneq 4536 . . . . . . . . . . . . . 14 (𝑡 = 𝑥 → {𝑡} = {𝑥})
2221breq2d 5052 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → (𝐴 <<s {𝑡} ↔ 𝐴 <<s {𝑥}))
2321breq1d 5050 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → ({𝑡} <<s 𝐵 ↔ {𝑥} <<s 𝐵))
2422, 23anbi12d 634 . . . . . . . . . . . 12 (𝑡 = 𝑥 → ((𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵) ↔ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)))
259adantr 484 . . . . . . . . . . . 12 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝐴 <<s {𝑥}) → 𝑥 No )
26 snex 5308 . . . . . . . . . . . . . . 15 {𝑥} ∈ V
2726a1i 11 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → {𝑥} ∈ V)
28 ssltex2 33637 . . . . . . . . . . . . . . 15 (𝐴 <<s 𝐵𝐵 ∈ V)
2928ad2antrr 726 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → 𝐵 ∈ V)
309snssd 4707 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → {𝑥} ⊆ No )
31 ssltss2 33639 . . . . . . . . . . . . . . 15 (𝐴 <<s 𝐵𝐵 No )
3231ad2antrr 726 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → 𝐵 No )
339adantr 484 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → 𝑥 No )
34 simpr 488 . . . . . . . . . . . . . . . . . . 19 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → 𝑋 = (𝐴 |s 𝐵))
35 simpl 486 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → 𝐴 <<s 𝐵)
3635scutcld 33652 . . . . . . . . . . . . . . . . . . 19 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (𝐴 |s 𝐵) ∈ No )
3734, 36eqeltrd 2834 . . . . . . . . . . . . . . . . . 18 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → 𝑋 No )
3837ad2antrr 726 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → 𝑋 No )
3932sselda 3887 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → 𝑏 No )
40 leftval 33702 . . . . . . . . . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (𝑥 ∈ ( L ‘𝑋) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋}))
43 rabid 3282 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋} ↔ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋))
4442, 43bitrdi 290 . . . . . . . . . . . . . . . . . . 19 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (𝑥 ∈ ( L ‘𝑋) ↔ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋)))
4544simplbda 503 . . . . . . . . . . . . . . . . . 18 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → 𝑥 <s 𝑋)
4645adantr 484 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → 𝑥 <s 𝑋)
47 simpllr 776 . . . . . . . . . . . . . . . . . 18 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → 𝑋 = (𝐴 |s 𝐵))
48 scutcut 33650 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
4948ad2antrr 726 . . . . . . . . . . . . . . . . . . . 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 1145 . . . . . . . . . . . . . . . . . . 19 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → {(𝐴 |s 𝐵)} <<s 𝐵)
51 ovex 7215 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 |s 𝐵) ∈ V
5251snid 4562 . . . . . . . . . . . . . . . . . . . 20 (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)}
53 ssltsepc 33642 . . . . . . . . . . . . . . . . . . . 20 (({(𝐴 |s 𝐵)} <<s 𝐵 ∧ (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)} ∧ 𝑏𝐵) → (𝐴 |s 𝐵) <s 𝑏)
5452, 53mp3an2 1450 . . . . . . . . . . . . . . . . . . 19 (({(𝐴 |s 𝐵)} <<s 𝐵𝑏𝐵) → (𝐴 |s 𝐵) <s 𝑏)
5550, 54sylan 583 . . . . . . . . . . . . . . . . . 18 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → (𝐴 |s 𝐵) <s 𝑏)
5647, 55eqbrtrd 5062 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → 𝑋 <s 𝑏)
5733, 38, 39, 46, 56slttrd 33617 . . . . . . . . . . . . . . . 16 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑏𝐵) → 𝑥 <s 𝑏)
58573adant2 1132 . . . . . . . . . . . . . . 15 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑎 ∈ {𝑥} ∧ 𝑏𝐵) → 𝑥 <s 𝑏)
59 velsn 4542 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ {𝑥} ↔ 𝑎 = 𝑥)
60 breq1 5043 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → (𝑎 <s 𝑏𝑥 <s 𝑏))
6159, 60sylbi 220 . . . . . . . . . . . . . . . 16 (𝑎 ∈ {𝑥} → (𝑎 <s 𝑏𝑥 <s 𝑏))
62613ad2ant2 1135 . . . . . . . . . . . . . . 15 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑎 ∈ {𝑥} ∧ 𝑏𝐵) → (𝑎 <s 𝑏𝑥 <s 𝑏))
6358, 62mpbird 260 . . . . . . . . . . . . . 14 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑎 ∈ {𝑥} ∧ 𝑏𝐵) → 𝑎 <s 𝑏)
6427, 29, 30, 32, 63ssltd 33641 . . . . . . . . . . . . 13 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → {𝑥} <<s 𝐵)
6564anim1ci 619 . . . . . . . . . . . 12 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝐴 <<s {𝑥}) → (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))
6624, 25, 65elrabd 3595 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝐴 <<s {𝑥}) → 𝑥 ∈ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)})
67 fnfvima 7018 . . . . . . . . . . 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 1466 . . . . . . . . . 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 4861 . . . . . . . . . 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 3925 . . . . . . . 8 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝐴 <<s {𝑥}) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑥))
7216, 71mtand 816 . . . . . . 7 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ¬ 𝐴 <<s {𝑥})
73 ssltex1 33636 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐴 ∈ V)
7473ad3antrrr 730 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡) → 𝐴 ∈ V)
7574, 26jctir 524 . . . . . . . 8 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡) → (𝐴 ∈ V ∧ {𝑥} ∈ V))
76 ssltss1 33638 . . . . . . . . . 10 (𝐴 <<s 𝐵𝐴 No )
7776ad3antrrr 730 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡) → 𝐴 No )
789adantr 484 . . . . . . . . . 10 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡) → 𝑥 No )
7978snssd 4707 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡) → {𝑥} ⊆ No )
80 simpr 488 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡) → ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡)
8177, 79, 803jca 1129 . . . . . . . 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 33635 . . . . . . . 8 (𝐴 <<s {𝑥} ↔ ((𝐴 ∈ V ∧ {𝑥} ∈ V) ∧ (𝐴 No ∧ {𝑥} ⊆ No ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡)))
8375, 81, 82sylanbrc 586 . . . . . . 7 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡) → 𝐴 <<s {𝑥})
8472, 83mtand 816 . . . . . 6 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ¬ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡)
85 rexnal 3152 . . . . . . 7 (∃𝑡 ∈ {𝑥} ¬ ∀𝑦𝐴 𝑦 <s 𝑡 ↔ ¬ ∀𝑡 ∈ {𝑥}∀𝑦𝐴 𝑦 <s 𝑡)
86 ralcom 3259 . . . . . . 7 (∀𝑡 ∈ {𝑥}∀𝑦𝐴 𝑦 <s 𝑡 ↔ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡)
8785, 86xchbinx 337 . . . . . 6 (∃𝑡 ∈ {𝑥} ¬ ∀𝑦𝐴 𝑦 <s 𝑡 ↔ ¬ ∀𝑦𝐴𝑡 ∈ {𝑥}𝑦 <s 𝑡)
8884, 87sylibr 237 . . . . 5 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ∃𝑡 ∈ {𝑥} ¬ ∀𝑦𝐴 𝑦 <s 𝑡)
89 vex 3404 . . . . . 6 𝑥 ∈ V
90 breq2 5044 . . . . . . . 8 (𝑡 = 𝑥 → (𝑦 <s 𝑡𝑦 <s 𝑥))
9190ralbidv 3110 . . . . . . 7 (𝑡 = 𝑥 → (∀𝑦𝐴 𝑦 <s 𝑡 ↔ ∀𝑦𝐴 𝑦 <s 𝑥))
9291notbid 321 . . . . . 6 (𝑡 = 𝑥 → (¬ ∀𝑦𝐴 𝑦 <s 𝑡 ↔ ¬ ∀𝑦𝐴 𝑦 <s 𝑥))
9389, 92rexsn 4583 . . . . 5 (∃𝑡 ∈ {𝑥} ¬ ∀𝑦𝐴 𝑦 <s 𝑡 ↔ ¬ ∀𝑦𝐴 𝑦 <s 𝑥)
9488, 93sylib 221 . . . 4 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ¬ ∀𝑦𝐴 𝑦 <s 𝑥)
9576ad2antrr 726 . . . . . . . 8 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → 𝐴 No )
9695sselda 3887 . . . . . . 7 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑦𝐴) → 𝑦 No )
97 slenlt 33610 . . . . . . 7 ((𝑥 No 𝑦 No ) → (𝑥 ≤s 𝑦 ↔ ¬ 𝑦 <s 𝑥))
989, 96, 97syl2an2r 685 . . . . . 6 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) ∧ 𝑦𝐴) → (𝑥 ≤s 𝑦 ↔ ¬ 𝑦 <s 𝑥))
9998rexbidva 3207 . . . . 5 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → (∃𝑦𝐴 𝑥 ≤s 𝑦 ↔ ∃𝑦𝐴 ¬ 𝑦 <s 𝑥))
100 rexnal 3152 . . . . 5 (∃𝑦𝐴 ¬ 𝑦 <s 𝑥 ↔ ¬ ∀𝑦𝐴 𝑦 <s 𝑥)
10199, 100bitrdi 290 . . . 4 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → (∃𝑦𝐴 𝑥 ≤s 𝑦 ↔ ¬ ∀𝑦𝐴 𝑦 <s 𝑥))
10294, 101mpbird 260 . . 3 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑥 ∈ ( L ‘𝑋)) → ∃𝑦𝐴 𝑥 ≤s 𝑦)
103102ralrimiva 3097 . 2 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ∀𝑥 ∈ ( L ‘𝑋)∃𝑦𝐴 𝑥 ≤s 𝑦)
1041onssneli 6292 . . . . . . . . 9 (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑧) → ¬ ( bday 𝑧) ∈ ( bday ‘(𝐴 |s 𝐵)))
105 rightssold 33717 . . . . . . . . . . . . 13 ( R ‘𝑋) ⊆ ( O ‘( bday 𝑋))
106105a1i 11 . . . . . . . . . . . 12 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ( R ‘𝑋) ⊆ ( O ‘( bday 𝑋)))
107106sselda 3887 . . . . . . . . . . 11 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝑧 ∈ ( O ‘( bday 𝑋)))
108 rightssno 33719 . . . . . . . . . . . . . 14 ( R ‘𝑋) ⊆ No
109108a1i 11 . . . . . . . . . . . . 13 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ( R ‘𝑋) ⊆ No )
110109sselda 3887 . . . . . . . . . . . 12 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝑧 No )
111 oldbday 33736 . . . . . . . . . . . 12 ((( bday 𝑋) ∈ On ∧ 𝑧 No ) → (𝑧 ∈ ( O ‘( bday 𝑋)) ↔ ( bday 𝑧) ∈ ( bday 𝑋)))
1126, 110, 111sylancr 590 . . . . . . . . . . 11 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → (𝑧 ∈ ( O ‘( bday 𝑋)) ↔ ( bday 𝑧) ∈ ( bday 𝑋)))
113107, 112mpbid 235 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ( bday 𝑧) ∈ ( bday 𝑋))
114 simplr 769 . . . . . . . . . . 11 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝑋 = (𝐴 |s 𝐵))
115114fveq2d 6690 . . . . . . . . . 10 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ( bday 𝑋) = ( bday ‘(𝐴 |s 𝐵)))
116113, 115eleqtrd 2836 . . . . . . . . 9 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ( bday 𝑧) ∈ ( bday ‘(𝐴 |s 𝐵)))
117104, 116nsyl3 140 . . . . . . . 8 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ¬ ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑧))
11817ad3antrrr 730 . . . . . . . . 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 4536 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → {𝑡} = {𝑧})
120119breq2d 5052 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (𝐴 <<s {𝑡} ↔ 𝐴 <<s {𝑧}))
121119breq1d 5050 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → ({𝑡} <<s 𝐵 ↔ {𝑧} <<s 𝐵))
122120, 121anbi12d 634 . . . . . . . . . . . 12 (𝑡 = 𝑧 → ((𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵) ↔ (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵)))
123110adantr 484 . . . . . . . . . . . 12 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ {𝑧} <<s 𝐵) → 𝑧 No )
12473ad2antrr 726 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝐴 ∈ V)
125 snex 5308 . . . . . . . . . . . . . . 15 {𝑧} ∈ V
126125a1i 11 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → {𝑧} ∈ V)
12776ad2antrr 726 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝐴 No )
128110snssd 4707 . . . . . . . . . . . . . 14 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → {𝑧} ⊆ No )
129127sselda 3887 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑎 No )
13037ad2antrr 726 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑋 No )
131110adantr 484 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑧 No )
13248ad2antrr 726 . . . . . . . . . . . . . . . . . . . 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 1144 . . . . . . . . . . . . . . . . . . 19 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝐴 <<s {(𝐴 |s 𝐵)})
134 ssltsepc 33642 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 <<s {(𝐴 |s 𝐵)} ∧ 𝑎𝐴 ∧ (𝐴 |s 𝐵) ∈ {(𝐴 |s 𝐵)}) → 𝑎 <s (𝐴 |s 𝐵))
13552, 134mp3an3 1451 . . . . . . . . . . . . . . . . . . 19 ((𝐴 <<s {(𝐴 |s 𝐵)} ∧ 𝑎𝐴) → 𝑎 <s (𝐴 |s 𝐵))
136133, 135sylan 583 . . . . . . . . . . . . . . . . . 18 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑎 <s (𝐴 |s 𝐵))
137 simpllr 776 . . . . . . . . . . . . . . . . . 18 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑋 = (𝐴 |s 𝐵))
138136, 137breqtrrd 5068 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑎 <s 𝑋)
139 rightval 33703 . . . . . . . . . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (𝑧 ∈ ( R ‘𝑋) ↔ 𝑧 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧}))
142 rabid 3282 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ {𝑧 ∈ ( O ‘( bday 𝑋)) ∣ 𝑋 <s 𝑧} ↔ (𝑧 ∈ ( O ‘( bday 𝑋)) ∧ 𝑋 <s 𝑧))
143141, 142bitrdi 290 . . . . . . . . . . . . . . . . . . 19 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → (𝑧 ∈ ( R ‘𝑋) ↔ (𝑧 ∈ ( O ‘( bday 𝑋)) ∧ 𝑋 <s 𝑧)))
144143simplbda 503 . . . . . . . . . . . . . . . . . 18 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝑋 <s 𝑧)
145144adantr 484 . . . . . . . . . . . . . . . . 17 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑋 <s 𝑧)
146129, 130, 131, 138, 145slttrd 33617 . . . . . . . . . . . . . . . 16 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴) → 𝑎 <s 𝑧)
1471463adant3 1133 . . . . . . . . . . . . . . 15 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴𝑏 ∈ {𝑧}) → 𝑎 <s 𝑧)
148 velsn 4542 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ {𝑧} ↔ 𝑏 = 𝑧)
149 breq2 5044 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑧 → (𝑎 <s 𝑏𝑎 <s 𝑧))
150148, 149sylbi 220 . . . . . . . . . . . . . . . 16 (𝑏 ∈ {𝑧} → (𝑎 <s 𝑏𝑎 <s 𝑧))
1511503ad2ant3 1136 . . . . . . . . . . . . . . 15 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴𝑏 ∈ {𝑧}) → (𝑎 <s 𝑏𝑎 <s 𝑧))
152147, 151mpbird 260 . . . . . . . . . . . . . 14 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑎𝐴𝑏 ∈ {𝑧}) → 𝑎 <s 𝑏)
153124, 126, 127, 128, 152ssltd 33641 . . . . . . . . . . . . 13 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝐴 <<s {𝑧})
154153anim1i 618 . . . . . . . . . . . 12 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ {𝑧} <<s 𝐵) → (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵))
155122, 123, 154elrabd 3595 . . . . . . . . . . 11 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ {𝑧} <<s 𝐵) → 𝑧 ∈ {𝑡 No ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)})
156 fnfvima 7018 . . . . . . . . . . 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 1466 . . . . . . . . . 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 4861 . . . . . . . . . 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 3925 . . . . . . . 8 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ {𝑧} <<s 𝐵) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday 𝑧))
161117, 160mtand 816 . . . . . . 7 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ¬ {𝑧} <<s 𝐵)
16228ad3antrrr 730 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤) → 𝐵 ∈ V)
163162, 125jctil 523 . . . . . . . 8 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤) → ({𝑧} ∈ V ∧ 𝐵 ∈ V))
164128adantr 484 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤) → {𝑧} ⊆ No )
16531ad3antrrr 730 . . . . . . . . 9 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤) → 𝐵 No )
166 simpr 488 . . . . . . . . 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 1129 . . . . . . . 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 33635 . . . . . . . 8 ({𝑧} <<s 𝐵 ↔ (({𝑧} ∈ V ∧ 𝐵 ∈ V) ∧ ({𝑧} ⊆ No 𝐵 No ∧ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤)))
169163, 167, 168sylanbrc 586 . . . . . . 7 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤) → {𝑧} <<s 𝐵)
170161, 169mtand 816 . . . . . 6 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ¬ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤)
171 rexnal 3152 . . . . . 6 (∃𝑡 ∈ {𝑧} ¬ ∀𝑤𝐵 𝑡 <s 𝑤 ↔ ¬ ∀𝑡 ∈ {𝑧}∀𝑤𝐵 𝑡 <s 𝑤)
172170, 171sylibr 237 . . . . 5 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ∃𝑡 ∈ {𝑧} ¬ ∀𝑤𝐵 𝑡 <s 𝑤)
173 vex 3404 . . . . . 6 𝑧 ∈ V
174 breq1 5043 . . . . . . . 8 (𝑡 = 𝑧 → (𝑡 <s 𝑤𝑧 <s 𝑤))
175174ralbidv 3110 . . . . . . 7 (𝑡 = 𝑧 → (∀𝑤𝐵 𝑡 <s 𝑤 ↔ ∀𝑤𝐵 𝑧 <s 𝑤))
176175notbid 321 . . . . . 6 (𝑡 = 𝑧 → (¬ ∀𝑤𝐵 𝑡 <s 𝑤 ↔ ¬ ∀𝑤𝐵 𝑧 <s 𝑤))
177173, 176rexsn 4583 . . . . 5 (∃𝑡 ∈ {𝑧} ¬ ∀𝑤𝐵 𝑡 <s 𝑤 ↔ ¬ ∀𝑤𝐵 𝑧 <s 𝑤)
178172, 177sylib 221 . . . 4 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ¬ ∀𝑤𝐵 𝑧 <s 𝑤)
17931ad2antrr 726 . . . . . . . 8 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → 𝐵 No )
180179sselda 3887 . . . . . . 7 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑤𝐵) → 𝑤 No )
181110adantr 484 . . . . . . 7 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑤𝐵) → 𝑧 No )
182 slenlt 33610 . . . . . . 7 ((𝑤 No 𝑧 No ) → (𝑤 ≤s 𝑧 ↔ ¬ 𝑧 <s 𝑤))
183180, 181, 182syl2anc 587 . . . . . 6 ((((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) ∧ 𝑤𝐵) → (𝑤 ≤s 𝑧 ↔ ¬ 𝑧 <s 𝑤))
184183rexbidva 3207 . . . . 5 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → (∃𝑤𝐵 𝑤 ≤s 𝑧 ↔ ∃𝑤𝐵 ¬ 𝑧 <s 𝑤))
185 rexnal 3152 . . . . 5 (∃𝑤𝐵 ¬ 𝑧 <s 𝑤 ↔ ¬ ∀𝑤𝐵 𝑧 <s 𝑤)
186184, 185bitrdi 290 . . . 4 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → (∃𝑤𝐵 𝑤 ≤s 𝑧 ↔ ¬ ∀𝑤𝐵 𝑧 <s 𝑤))
187178, 186mpbird 260 . . 3 (((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) ∧ 𝑧 ∈ ( R ‘𝑋)) → ∃𝑤𝐵 𝑤 ≤s 𝑧)
188187ralrimiva 3097 . 2 ((𝐴 <<s 𝐵𝑋 = (𝐴 |s 𝐵)) → ∀𝑧 ∈ ( R ‘𝑋)∃𝑤𝐵 𝑤 ≤s 𝑧)
189103, 188jca 515 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  wrex 3055  {crab 3058  Vcvv 3400  wss 3853  {csn 4526   cint 4846   class class class wbr 5040  cima 5538  Oncon0 6182   Fn wfn 6344  cfv 6349  (class class class)co 7182   No csur 33498   <s cslt 33499   bday cbday 33500   ≤s csle 33602   <<s csslt 33630   |s cscut 33632   O cold 33682   L cleft 33684   R cright 33685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-int 4847  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6185  df-on 6186  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7139  df-ov 7185  df-oprab 7186  df-mpo 7187  df-wrecs 7988  df-recs 8049  df-1o 8143  df-2o 8144  df-no 33501  df-slt 33502  df-bday 33503  df-sle 33603  df-sslt 33631  df-scut 33633  df-made 33686  df-old 33687  df-left 33689  df-right 33690
This theorem is referenced by: (None)
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