| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ssltleft |
⊢ ( 𝑋 ∈ No → ( L ‘ 𝑋 ) <<s { 𝑋 } ) |
| 2 |
1
|
adantl |
⊢ ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) → ( L ‘ 𝑋 ) <<s { 𝑋 } ) |
| 3 |
|
ssltright |
⊢ ( 𝑋 ∈ No → { 𝑋 } <<s ( R ‘ 𝑋 ) ) |
| 4 |
3
|
adantl |
⊢ ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) → { 𝑋 } <<s ( R ‘ 𝑋 ) ) |
| 5 |
|
fveq2 |
⊢ ( 𝑋 = 𝑤 → ( bday ‘ 𝑋 ) = ( bday ‘ 𝑤 ) ) |
| 6 |
|
eqimss |
⊢ ( ( bday ‘ 𝑋 ) = ( bday ‘ 𝑤 ) → ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) |
| 7 |
5 6
|
syl |
⊢ ( 𝑋 = 𝑤 → ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) |
| 8 |
7
|
a1i |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ( L ‘ 𝑋 ) <<s { 𝑤 } ∧ { 𝑤 } <<s ( R ‘ 𝑋 ) ) ) ) → ( 𝑋 = 𝑤 → ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) ) |
| 9 |
|
ssltsep |
⊢ ( ( L ‘ 𝑋 ) <<s { 𝑤 } → ∀ 𝑥 ∈ ( L ‘ 𝑋 ) ∀ 𝑦 ∈ { 𝑤 } 𝑥 <s 𝑦 ) |
| 10 |
|
vex |
⊢ 𝑤 ∈ V |
| 11 |
|
breq2 |
⊢ ( 𝑦 = 𝑤 → ( 𝑥 <s 𝑦 ↔ 𝑥 <s 𝑤 ) ) |
| 12 |
10 11
|
ralsn |
⊢ ( ∀ 𝑦 ∈ { 𝑤 } 𝑥 <s 𝑦 ↔ 𝑥 <s 𝑤 ) |
| 13 |
12
|
ralbii |
⊢ ( ∀ 𝑥 ∈ ( L ‘ 𝑋 ) ∀ 𝑦 ∈ { 𝑤 } 𝑥 <s 𝑦 ↔ ∀ 𝑥 ∈ ( L ‘ 𝑋 ) 𝑥 <s 𝑤 ) |
| 14 |
9 13
|
sylib |
⊢ ( ( L ‘ 𝑋 ) <<s { 𝑤 } → ∀ 𝑥 ∈ ( L ‘ 𝑋 ) 𝑥 <s 𝑤 ) |
| 15 |
|
ssltsep |
⊢ ( { 𝑤 } <<s ( R ‘ 𝑋 ) → ∀ 𝑦 ∈ { 𝑤 } ∀ 𝑥 ∈ ( R ‘ 𝑋 ) 𝑦 <s 𝑥 ) |
| 16 |
|
breq1 |
⊢ ( 𝑦 = 𝑤 → ( 𝑦 <s 𝑥 ↔ 𝑤 <s 𝑥 ) ) |
| 17 |
16
|
ralbidv |
⊢ ( 𝑦 = 𝑤 → ( ∀ 𝑥 ∈ ( R ‘ 𝑋 ) 𝑦 <s 𝑥 ↔ ∀ 𝑥 ∈ ( R ‘ 𝑋 ) 𝑤 <s 𝑥 ) ) |
| 18 |
10 17
|
ralsn |
⊢ ( ∀ 𝑦 ∈ { 𝑤 } ∀ 𝑥 ∈ ( R ‘ 𝑋 ) 𝑦 <s 𝑥 ↔ ∀ 𝑥 ∈ ( R ‘ 𝑋 ) 𝑤 <s 𝑥 ) |
| 19 |
15 18
|
sylib |
⊢ ( { 𝑤 } <<s ( R ‘ 𝑋 ) → ∀ 𝑥 ∈ ( R ‘ 𝑋 ) 𝑤 <s 𝑥 ) |
| 20 |
14 19
|
anim12i |
⊢ ( ( ( L ‘ 𝑋 ) <<s { 𝑤 } ∧ { 𝑤 } <<s ( R ‘ 𝑋 ) ) → ( ∀ 𝑥 ∈ ( L ‘ 𝑋 ) 𝑥 <s 𝑤 ∧ ∀ 𝑥 ∈ ( R ‘ 𝑋 ) 𝑤 <s 𝑥 ) ) |
| 21 |
|
leftval |
⊢ ( 𝑋 ∈ No → ( L ‘ 𝑋 ) = { 𝑧 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ∣ 𝑧 <s 𝑋 } ) |
| 22 |
21
|
raleqdv |
⊢ ( 𝑋 ∈ No → ( ∀ 𝑥 ∈ ( L ‘ 𝑋 ) 𝑥 <s 𝑤 ↔ ∀ 𝑥 ∈ { 𝑧 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ∣ 𝑧 <s 𝑋 } 𝑥 <s 𝑤 ) ) |
| 23 |
|
rightval |
⊢ ( 𝑋 ∈ No → ( R ‘ 𝑋 ) = { 𝑧 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ∣ 𝑋 <s 𝑧 } ) |
| 24 |
23
|
raleqdv |
⊢ ( 𝑋 ∈ No → ( ∀ 𝑥 ∈ ( R ‘ 𝑋 ) 𝑤 <s 𝑥 ↔ ∀ 𝑥 ∈ { 𝑧 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ∣ 𝑋 <s 𝑧 } 𝑤 <s 𝑥 ) ) |
| 25 |
22 24
|
anbi12d |
⊢ ( 𝑋 ∈ No → ( ( ∀ 𝑥 ∈ ( L ‘ 𝑋 ) 𝑥 <s 𝑤 ∧ ∀ 𝑥 ∈ ( R ‘ 𝑋 ) 𝑤 <s 𝑥 ) ↔ ( ∀ 𝑥 ∈ { 𝑧 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ∣ 𝑧 <s 𝑋 } 𝑥 <s 𝑤 ∧ ∀ 𝑥 ∈ { 𝑧 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ∣ 𝑋 <s 𝑧 } 𝑤 <s 𝑥 ) ) ) |
| 26 |
|
breq1 |
⊢ ( 𝑧 = 𝑥 → ( 𝑧 <s 𝑋 ↔ 𝑥 <s 𝑋 ) ) |
| 27 |
26
|
ralrab |
⊢ ( ∀ 𝑥 ∈ { 𝑧 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ∣ 𝑧 <s 𝑋 } 𝑥 <s 𝑤 ↔ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ) |
| 28 |
|
breq2 |
⊢ ( 𝑧 = 𝑥 → ( 𝑋 <s 𝑧 ↔ 𝑋 <s 𝑥 ) ) |
| 29 |
28
|
ralrab |
⊢ ( ∀ 𝑥 ∈ { 𝑧 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ∣ 𝑋 <s 𝑧 } 𝑤 <s 𝑥 ↔ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) |
| 30 |
27 29
|
anbi12i |
⊢ ( ( ∀ 𝑥 ∈ { 𝑧 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ∣ 𝑧 <s 𝑋 } 𝑥 <s 𝑤 ∧ ∀ 𝑥 ∈ { 𝑧 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ∣ 𝑋 <s 𝑧 } 𝑤 <s 𝑥 ) ↔ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) |
| 31 |
25 30
|
bitrdi |
⊢ ( 𝑋 ∈ No → ( ( ∀ 𝑥 ∈ ( L ‘ 𝑋 ) 𝑥 <s 𝑤 ∧ ∀ 𝑥 ∈ ( R ‘ 𝑋 ) 𝑤 <s 𝑥 ) ↔ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) |
| 32 |
31
|
ad2antlr |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ 𝑤 ∈ No ) → ( ( ∀ 𝑥 ∈ ( L ‘ 𝑋 ) 𝑥 <s 𝑤 ∧ ∀ 𝑥 ∈ ( R ‘ 𝑋 ) 𝑤 <s 𝑥 ) ↔ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) |
| 33 |
|
simplrl |
⊢ ( ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) ∧ 𝑋 ≠ 𝑤 ) → 𝑤 ∈ No ) |
| 34 |
|
sltirr |
⊢ ( 𝑤 ∈ No → ¬ 𝑤 <s 𝑤 ) |
| 35 |
33 34
|
syl |
⊢ ( ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) ∧ 𝑋 ≠ 𝑤 ) → ¬ 𝑤 <s 𝑤 ) |
| 36 |
|
bdayelon |
⊢ ( bday ‘ 𝑋 ) ∈ On |
| 37 |
|
bdayelon |
⊢ ( bday ‘ 𝑤 ) ∈ On |
| 38 |
|
ontri1 |
⊢ ( ( ( bday ‘ 𝑋 ) ∈ On ∧ ( bday ‘ 𝑤 ) ∈ On ) → ( ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ↔ ¬ ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ) ) |
| 39 |
36 37 38
|
mp2an |
⊢ ( ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ↔ ¬ ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ) |
| 40 |
39
|
con2bii |
⊢ ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ↔ ¬ ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) |
| 41 |
|
simplll |
⊢ ( ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) ∧ 𝑋 ≠ 𝑤 ) → ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) |
| 42 |
|
madebdaylemold |
⊢ ( ( ( bday ‘ 𝑋 ) ∈ On ∧ ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) → 𝑤 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ) ) |
| 43 |
36 41 33 42
|
mp3an2i |
⊢ ( ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) ∧ 𝑋 ≠ 𝑤 ) → ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) → 𝑤 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ) ) |
| 44 |
|
slttrine |
⊢ ( ( 𝑋 ∈ No ∧ 𝑤 ∈ No ) → ( 𝑋 ≠ 𝑤 ↔ ( 𝑋 <s 𝑤 ∨ 𝑤 <s 𝑋 ) ) ) |
| 45 |
44
|
ad2ant2lr |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) → ( 𝑋 ≠ 𝑤 ↔ ( 𝑋 <s 𝑤 ∨ 𝑤 <s 𝑋 ) ) ) |
| 46 |
|
simprrr |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) → ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) |
| 47 |
|
breq2 |
⊢ ( 𝑥 = 𝑤 → ( 𝑋 <s 𝑥 ↔ 𝑋 <s 𝑤 ) ) |
| 48 |
|
breq2 |
⊢ ( 𝑥 = 𝑤 → ( 𝑤 <s 𝑥 ↔ 𝑤 <s 𝑤 ) ) |
| 49 |
47 48
|
imbi12d |
⊢ ( 𝑥 = 𝑤 → ( ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ↔ ( 𝑋 <s 𝑤 → 𝑤 <s 𝑤 ) ) ) |
| 50 |
49
|
rspccv |
⊢ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) → ( 𝑤 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) → ( 𝑋 <s 𝑤 → 𝑤 <s 𝑤 ) ) ) |
| 51 |
46 50
|
syl |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) → ( 𝑤 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) → ( 𝑋 <s 𝑤 → 𝑤 <s 𝑤 ) ) ) |
| 52 |
51
|
com23 |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) → ( 𝑋 <s 𝑤 → ( 𝑤 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) → 𝑤 <s 𝑤 ) ) ) |
| 53 |
|
simprrl |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) → ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ) |
| 54 |
|
breq1 |
⊢ ( 𝑥 = 𝑤 → ( 𝑥 <s 𝑋 ↔ 𝑤 <s 𝑋 ) ) |
| 55 |
|
breq1 |
⊢ ( 𝑥 = 𝑤 → ( 𝑥 <s 𝑤 ↔ 𝑤 <s 𝑤 ) ) |
| 56 |
54 55
|
imbi12d |
⊢ ( 𝑥 = 𝑤 → ( ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ↔ ( 𝑤 <s 𝑋 → 𝑤 <s 𝑤 ) ) ) |
| 57 |
56
|
rspccv |
⊢ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) → ( 𝑤 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) → ( 𝑤 <s 𝑋 → 𝑤 <s 𝑤 ) ) ) |
| 58 |
53 57
|
syl |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) → ( 𝑤 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) → ( 𝑤 <s 𝑋 → 𝑤 <s 𝑤 ) ) ) |
| 59 |
58
|
com23 |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) → ( 𝑤 <s 𝑋 → ( 𝑤 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) → 𝑤 <s 𝑤 ) ) ) |
| 60 |
52 59
|
jaod |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) → ( ( 𝑋 <s 𝑤 ∨ 𝑤 <s 𝑋 ) → ( 𝑤 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) → 𝑤 <s 𝑤 ) ) ) |
| 61 |
45 60
|
sylbid |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) → ( 𝑋 ≠ 𝑤 → ( 𝑤 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) → 𝑤 <s 𝑤 ) ) ) |
| 62 |
61
|
imp |
⊢ ( ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) ∧ 𝑋 ≠ 𝑤 ) → ( 𝑤 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) → 𝑤 <s 𝑤 ) ) |
| 63 |
43 62
|
syld |
⊢ ( ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) ∧ 𝑋 ≠ 𝑤 ) → ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) → 𝑤 <s 𝑤 ) ) |
| 64 |
40 63
|
syl5bir |
⊢ ( ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) ∧ 𝑋 ≠ 𝑤 ) → ( ¬ ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) → 𝑤 <s 𝑤 ) ) |
| 65 |
35 64
|
mt3d |
⊢ ( ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) ∧ 𝑋 ≠ 𝑤 ) → ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) |
| 66 |
65
|
ex |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) ) ) → ( 𝑋 ≠ 𝑤 → ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) ) |
| 67 |
66
|
expr |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ 𝑤 ∈ No ) → ( ( ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑥 <s 𝑋 → 𝑥 <s 𝑤 ) ∧ ∀ 𝑥 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ( 𝑋 <s 𝑥 → 𝑤 <s 𝑥 ) ) → ( 𝑋 ≠ 𝑤 → ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) ) ) |
| 68 |
32 67
|
sylbid |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ 𝑤 ∈ No ) → ( ( ∀ 𝑥 ∈ ( L ‘ 𝑋 ) 𝑥 <s 𝑤 ∧ ∀ 𝑥 ∈ ( R ‘ 𝑋 ) 𝑤 <s 𝑥 ) → ( 𝑋 ≠ 𝑤 → ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) ) ) |
| 69 |
68
|
impr |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ∀ 𝑥 ∈ ( L ‘ 𝑋 ) 𝑥 <s 𝑤 ∧ ∀ 𝑥 ∈ ( R ‘ 𝑋 ) 𝑤 <s 𝑥 ) ) ) → ( 𝑋 ≠ 𝑤 → ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) ) |
| 70 |
20 69
|
sylanr2 |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ( L ‘ 𝑋 ) <<s { 𝑤 } ∧ { 𝑤 } <<s ( R ‘ 𝑋 ) ) ) ) → ( 𝑋 ≠ 𝑤 → ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) ) |
| 71 |
8 70
|
pm2.61dne |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ ( 𝑤 ∈ No ∧ ( ( L ‘ 𝑋 ) <<s { 𝑤 } ∧ { 𝑤 } <<s ( R ‘ 𝑋 ) ) ) ) → ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) |
| 72 |
71
|
expr |
⊢ ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) ∧ 𝑤 ∈ No ) → ( ( ( L ‘ 𝑋 ) <<s { 𝑤 } ∧ { 𝑤 } <<s ( R ‘ 𝑋 ) ) → ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) ) |
| 73 |
72
|
ralrimiva |
⊢ ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) → ∀ 𝑤 ∈ No ( ( ( L ‘ 𝑋 ) <<s { 𝑤 } ∧ { 𝑤 } <<s ( R ‘ 𝑋 ) ) → ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) ) |
| 74 |
|
bdayfn |
⊢ bday Fn No |
| 75 |
|
ssrab2 |
⊢ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ⊆ No |
| 76 |
|
fnssintima |
⊢ ( ( bday Fn No ∧ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ⊆ No ) → ( ( bday ‘ 𝑋 ) ⊆ ∩ ( bday “ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ) ↔ ∀ 𝑤 ∈ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) ) |
| 77 |
74 75 76
|
mp2an |
⊢ ( ( bday ‘ 𝑋 ) ⊆ ∩ ( bday “ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ) ↔ ∀ 𝑤 ∈ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) |
| 78 |
|
sneq |
⊢ ( 𝑧 = 𝑤 → { 𝑧 } = { 𝑤 } ) |
| 79 |
78
|
breq2d |
⊢ ( 𝑧 = 𝑤 → ( ( L ‘ 𝑋 ) <<s { 𝑧 } ↔ ( L ‘ 𝑋 ) <<s { 𝑤 } ) ) |
| 80 |
78
|
breq1d |
⊢ ( 𝑧 = 𝑤 → ( { 𝑧 } <<s ( R ‘ 𝑋 ) ↔ { 𝑤 } <<s ( R ‘ 𝑋 ) ) ) |
| 81 |
79 80
|
anbi12d |
⊢ ( 𝑧 = 𝑤 → ( ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) ↔ ( ( L ‘ 𝑋 ) <<s { 𝑤 } ∧ { 𝑤 } <<s ( R ‘ 𝑋 ) ) ) ) |
| 82 |
81
|
ralrab |
⊢ ( ∀ 𝑤 ∈ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ↔ ∀ 𝑤 ∈ No ( ( ( L ‘ 𝑋 ) <<s { 𝑤 } ∧ { 𝑤 } <<s ( R ‘ 𝑋 ) ) → ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) ) |
| 83 |
77 82
|
bitri |
⊢ ( ( bday ‘ 𝑋 ) ⊆ ∩ ( bday “ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ) ↔ ∀ 𝑤 ∈ No ( ( ( L ‘ 𝑋 ) <<s { 𝑤 } ∧ { 𝑤 } <<s ( R ‘ 𝑋 ) ) → ( bday ‘ 𝑋 ) ⊆ ( bday ‘ 𝑤 ) ) ) |
| 84 |
73 83
|
sylibr |
⊢ ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) → ( bday ‘ 𝑋 ) ⊆ ∩ ( bday “ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ) ) |
| 85 |
|
sneq |
⊢ ( 𝑧 = 𝑋 → { 𝑧 } = { 𝑋 } ) |
| 86 |
85
|
breq2d |
⊢ ( 𝑧 = 𝑋 → ( ( L ‘ 𝑋 ) <<s { 𝑧 } ↔ ( L ‘ 𝑋 ) <<s { 𝑋 } ) ) |
| 87 |
85
|
breq1d |
⊢ ( 𝑧 = 𝑋 → ( { 𝑧 } <<s ( R ‘ 𝑋 ) ↔ { 𝑋 } <<s ( R ‘ 𝑋 ) ) ) |
| 88 |
86 87
|
anbi12d |
⊢ ( 𝑧 = 𝑋 → ( ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) ↔ ( ( L ‘ 𝑋 ) <<s { 𝑋 } ∧ { 𝑋 } <<s ( R ‘ 𝑋 ) ) ) ) |
| 89 |
|
simpr |
⊢ ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) → 𝑋 ∈ No ) |
| 90 |
2 4
|
jca |
⊢ ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) → ( ( L ‘ 𝑋 ) <<s { 𝑋 } ∧ { 𝑋 } <<s ( R ‘ 𝑋 ) ) ) |
| 91 |
88 89 90
|
elrabd |
⊢ ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) → 𝑋 ∈ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ) |
| 92 |
|
fnfvima |
⊢ ( ( bday Fn No ∧ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ⊆ No ∧ 𝑋 ∈ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ) → ( bday ‘ 𝑋 ) ∈ ( bday “ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ) ) |
| 93 |
74 75 91 92
|
mp3an12i |
⊢ ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) → ( bday ‘ 𝑋 ) ∈ ( bday “ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ) ) |
| 94 |
|
intss1 |
⊢ ( ( bday ‘ 𝑋 ) ∈ ( bday “ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ) → ∩ ( bday “ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ) ⊆ ( bday ‘ 𝑋 ) ) |
| 95 |
93 94
|
syl |
⊢ ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) → ∩ ( bday “ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ) ⊆ ( bday ‘ 𝑋 ) ) |
| 96 |
84 95
|
eqssd |
⊢ ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) → ( bday ‘ 𝑋 ) = ∩ ( bday “ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ) ) |
| 97 |
|
lltropt |
⊢ ( 𝑋 ∈ No → ( L ‘ 𝑋 ) <<s ( R ‘ 𝑋 ) ) |
| 98 |
|
eqscut |
⊢ ( ( ( L ‘ 𝑋 ) <<s ( R ‘ 𝑋 ) ∧ 𝑋 ∈ No ) → ( ( ( L ‘ 𝑋 ) |s ( R ‘ 𝑋 ) ) = 𝑋 ↔ ( ( L ‘ 𝑋 ) <<s { 𝑋 } ∧ { 𝑋 } <<s ( R ‘ 𝑋 ) ∧ ( bday ‘ 𝑋 ) = ∩ ( bday “ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ) ) ) ) |
| 99 |
97 98
|
mpancom |
⊢ ( 𝑋 ∈ No → ( ( ( L ‘ 𝑋 ) |s ( R ‘ 𝑋 ) ) = 𝑋 ↔ ( ( L ‘ 𝑋 ) <<s { 𝑋 } ∧ { 𝑋 } <<s ( R ‘ 𝑋 ) ∧ ( bday ‘ 𝑋 ) = ∩ ( bday “ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ) ) ) ) |
| 100 |
99
|
adantl |
⊢ ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) → ( ( ( L ‘ 𝑋 ) |s ( R ‘ 𝑋 ) ) = 𝑋 ↔ ( ( L ‘ 𝑋 ) <<s { 𝑋 } ∧ { 𝑋 } <<s ( R ‘ 𝑋 ) ∧ ( bday ‘ 𝑋 ) = ∩ ( bday “ { 𝑧 ∈ No ∣ ( ( L ‘ 𝑋 ) <<s { 𝑧 } ∧ { 𝑧 } <<s ( R ‘ 𝑋 ) ) } ) ) ) ) |
| 101 |
2 4 96 100
|
mpbir3and |
⊢ ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑋 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑋 ∈ No ) → ( ( L ‘ 𝑋 ) |s ( R ‘ 𝑋 ) ) = 𝑋 ) |