Step |
Hyp |
Ref |
Expression |
1 |
|
breq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 <s 𝑧 ↔ 𝑋 <s 𝑧 ) ) |
2 |
1
|
anbi1d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦 ) ↔ ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦 ) ) ) |
3 |
2
|
imbi1d |
⊢ ( 𝑥 = 𝑋 → ( ( ( 𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) ↔ ( ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) ) ) |
4 |
3
|
ralbidv |
⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑧 ∈ No ( ( 𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) ↔ ∀ 𝑧 ∈ No ( ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) ) ) |
5 |
|
breq2 |
⊢ ( 𝑦 = 𝑌 → ( 𝑧 <s 𝑦 ↔ 𝑧 <s 𝑌 ) ) |
6 |
5
|
anbi2d |
⊢ ( 𝑦 = 𝑌 → ( ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦 ) ↔ ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌 ) ) ) |
7 |
6
|
imbi1d |
⊢ ( 𝑦 = 𝑌 → ( ( ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) ↔ ( ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌 ) → 𝑧 ∈ 𝐴 ) ) ) |
8 |
7
|
ralbidv |
⊢ ( 𝑦 = 𝑌 → ( ∀ 𝑧 ∈ No ( ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) ↔ ∀ 𝑧 ∈ No ( ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌 ) → 𝑧 ∈ 𝐴 ) ) ) |
9 |
4 8
|
rspc2v |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ No ( ( 𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) → ∀ 𝑧 ∈ No ( ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌 ) → 𝑧 ∈ 𝐴 ) ) ) |
10 |
|
breq2 |
⊢ ( 𝑧 = 𝑤 → ( 𝑋 <s 𝑧 ↔ 𝑋 <s 𝑤 ) ) |
11 |
|
breq1 |
⊢ ( 𝑧 = 𝑤 → ( 𝑧 <s 𝑌 ↔ 𝑤 <s 𝑌 ) ) |
12 |
10 11
|
anbi12d |
⊢ ( 𝑧 = 𝑤 → ( ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌 ) ↔ ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) ) ) |
13 |
|
eleq1w |
⊢ ( 𝑧 = 𝑤 → ( 𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴 ) ) |
14 |
12 13
|
imbi12d |
⊢ ( 𝑧 = 𝑤 → ( ( ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌 ) → 𝑧 ∈ 𝐴 ) ↔ ( ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) → 𝑤 ∈ 𝐴 ) ) ) |
15 |
14
|
rspcv |
⊢ ( 𝑤 ∈ No → ( ∀ 𝑧 ∈ No ( ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌 ) → 𝑧 ∈ 𝐴 ) → ( ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) → 𝑤 ∈ 𝐴 ) ) ) |
16 |
|
bdaydm |
⊢ dom bday = No |
17 |
16
|
sseq2i |
⊢ ( 𝐴 ⊆ dom bday ↔ 𝐴 ⊆ No ) |
18 |
|
bdayfun |
⊢ Fun bday |
19 |
|
funfvima2 |
⊢ ( ( Fun bday ∧ 𝐴 ⊆ dom bday ) → ( 𝑤 ∈ 𝐴 → ( bday ‘ 𝑤 ) ∈ ( bday “ 𝐴 ) ) ) |
20 |
18 19
|
mpan |
⊢ ( 𝐴 ⊆ dom bday → ( 𝑤 ∈ 𝐴 → ( bday ‘ 𝑤 ) ∈ ( bday “ 𝐴 ) ) ) |
21 |
17 20
|
sylbir |
⊢ ( 𝐴 ⊆ No → ( 𝑤 ∈ 𝐴 → ( bday ‘ 𝑤 ) ∈ ( bday “ 𝐴 ) ) ) |
22 |
21
|
imp |
⊢ ( ( 𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴 ) → ( bday ‘ 𝑤 ) ∈ ( bday “ 𝐴 ) ) |
23 |
|
intss1 |
⊢ ( ( bday ‘ 𝑤 ) ∈ ( bday “ 𝐴 ) → ∩ ( bday “ 𝐴 ) ⊆ ( bday ‘ 𝑤 ) ) |
24 |
22 23
|
syl |
⊢ ( ( 𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴 ) → ∩ ( bday “ 𝐴 ) ⊆ ( bday ‘ 𝑤 ) ) |
25 |
|
imassrn |
⊢ ( bday “ 𝐴 ) ⊆ ran bday |
26 |
|
bdayrn |
⊢ ran bday = On |
27 |
25 26
|
sseqtri |
⊢ ( bday “ 𝐴 ) ⊆ On |
28 |
22
|
ne0d |
⊢ ( ( 𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴 ) → ( bday “ 𝐴 ) ≠ ∅ ) |
29 |
|
oninton |
⊢ ( ( ( bday “ 𝐴 ) ⊆ On ∧ ( bday “ 𝐴 ) ≠ ∅ ) → ∩ ( bday “ 𝐴 ) ∈ On ) |
30 |
27 28 29
|
sylancr |
⊢ ( ( 𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴 ) → ∩ ( bday “ 𝐴 ) ∈ On ) |
31 |
|
bdayelon |
⊢ ( bday ‘ 𝑤 ) ∈ On |
32 |
|
ontri1 |
⊢ ( ( ∩ ( bday “ 𝐴 ) ∈ On ∧ ( bday ‘ 𝑤 ) ∈ On ) → ( ∩ ( bday “ 𝐴 ) ⊆ ( bday ‘ 𝑤 ) ↔ ¬ ( bday ‘ 𝑤 ) ∈ ∩ ( bday “ 𝐴 ) ) ) |
33 |
30 31 32
|
sylancl |
⊢ ( ( 𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴 ) → ( ∩ ( bday “ 𝐴 ) ⊆ ( bday ‘ 𝑤 ) ↔ ¬ ( bday ‘ 𝑤 ) ∈ ∩ ( bday “ 𝐴 ) ) ) |
34 |
24 33
|
mpbid |
⊢ ( ( 𝐴 ⊆ No ∧ 𝑤 ∈ 𝐴 ) → ¬ ( bday ‘ 𝑤 ) ∈ ∩ ( bday “ 𝐴 ) ) |
35 |
34
|
ex |
⊢ ( 𝐴 ⊆ No → ( 𝑤 ∈ 𝐴 → ¬ ( bday ‘ 𝑤 ) ∈ ∩ ( bday “ 𝐴 ) ) ) |
36 |
|
eleq2 |
⊢ ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) → ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ↔ ( bday ‘ 𝑤 ) ∈ ∩ ( bday “ 𝐴 ) ) ) |
37 |
36
|
notbid |
⊢ ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) → ( ¬ ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ↔ ¬ ( bday ‘ 𝑤 ) ∈ ∩ ( bday “ 𝐴 ) ) ) |
38 |
37
|
biimprcd |
⊢ ( ¬ ( bday ‘ 𝑤 ) ∈ ∩ ( bday “ 𝐴 ) → ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) → ¬ ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ) ) |
39 |
35 38
|
syl6 |
⊢ ( 𝐴 ⊆ No → ( 𝑤 ∈ 𝐴 → ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) → ¬ ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ) ) ) |
40 |
39
|
com3l |
⊢ ( 𝑤 ∈ 𝐴 → ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) → ( 𝐴 ⊆ No → ¬ ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ) ) ) |
41 |
40
|
adantrd |
⊢ ( 𝑤 ∈ 𝐴 → ( ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) → ( 𝐴 ⊆ No → ¬ ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ) ) ) |
42 |
15 41
|
syl8 |
⊢ ( 𝑤 ∈ No → ( ∀ 𝑧 ∈ No ( ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌 ) → 𝑧 ∈ 𝐴 ) → ( ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) → ( ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) → ( 𝐴 ⊆ No → ¬ ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ) ) ) ) ) |
43 |
42
|
com35 |
⊢ ( 𝑤 ∈ No → ( ∀ 𝑧 ∈ No ( ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌 ) → 𝑧 ∈ 𝐴 ) → ( 𝐴 ⊆ No → ( ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) → ( ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) → ¬ ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ) ) ) ) ) |
44 |
43
|
com4l |
⊢ ( ∀ 𝑧 ∈ No ( ( 𝑋 <s 𝑧 ∧ 𝑧 <s 𝑌 ) → 𝑧 ∈ 𝐴 ) → ( 𝐴 ⊆ No → ( ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) → ( 𝑤 ∈ No → ( ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) → ¬ ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ) ) ) ) ) |
45 |
9 44
|
syl6 |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ No ( ( 𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) → ( 𝐴 ⊆ No → ( ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) → ( 𝑤 ∈ No → ( ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) → ¬ ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ) ) ) ) ) ) |
46 |
45
|
com3l |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ No ( ( 𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) → ( 𝐴 ⊆ No → ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) → ( 𝑤 ∈ No → ( ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) → ¬ ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ) ) ) ) ) ) |
47 |
46
|
impcom |
⊢ ( ( 𝐴 ⊆ No ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ No ( ( 𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) ) → ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) → ( 𝑤 ∈ No → ( ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) → ¬ ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ) ) ) ) ) |
48 |
47
|
imp42 |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ No ( ( 𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ∧ ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) ) ) ∧ 𝑤 ∈ No ) → ( ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) → ¬ ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ) ) |
49 |
48
|
con2d |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ No ( ( 𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ∧ ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) ) ) ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) → ¬ ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) ) ) |
50 |
|
3anass |
⊢ ( ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) ↔ ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ∧ ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) ) ) |
51 |
50
|
notbii |
⊢ ( ¬ ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) ↔ ¬ ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ∧ ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) ) ) |
52 |
|
imnan |
⊢ ( ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) → ¬ ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) ) ↔ ¬ ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ∧ ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) ) ) |
53 |
51 52
|
bitr4i |
⊢ ( ¬ ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) ↔ ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) → ¬ ( 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) ) ) |
54 |
49 53
|
sylibr |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ No ( ( 𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ∧ ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) ) ) ∧ 𝑤 ∈ No ) → ¬ ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) ) |
55 |
54
|
nrexdv |
⊢ ( ( ( 𝐴 ⊆ No ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ No ( ( 𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ∧ ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) ) ) → ¬ ∃ 𝑤 ∈ No ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) ) |
56 |
|
ssel |
⊢ ( 𝐴 ⊆ No → ( 𝑋 ∈ 𝐴 → 𝑋 ∈ No ) ) |
57 |
|
ssel |
⊢ ( 𝐴 ⊆ No → ( 𝑌 ∈ 𝐴 → 𝑌 ∈ No ) ) |
58 |
56 57
|
anim12d |
⊢ ( 𝐴 ⊆ No → ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 ∈ No ∧ 𝑌 ∈ No ) ) ) |
59 |
58
|
imp |
⊢ ( ( 𝐴 ⊆ No ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) → ( 𝑋 ∈ No ∧ 𝑌 ∈ No ) ) |
60 |
|
eqtr3 |
⊢ ( ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) → ( bday ‘ 𝑋 ) = ( bday ‘ 𝑌 ) ) |
61 |
59 60
|
anim12i |
⊢ ( ( ( 𝐴 ⊆ No ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) ∧ ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) ) → ( ( 𝑋 ∈ No ∧ 𝑌 ∈ No ) ∧ ( bday ‘ 𝑋 ) = ( bday ‘ 𝑌 ) ) ) |
62 |
61
|
anasss |
⊢ ( ( 𝐴 ⊆ No ∧ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ∧ ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) ) ) → ( ( 𝑋 ∈ No ∧ 𝑌 ∈ No ) ∧ ( bday ‘ 𝑋 ) = ( bday ‘ 𝑌 ) ) ) |
63 |
62
|
adantlr |
⊢ ( ( ( 𝐴 ⊆ No ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ No ( ( 𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ∧ ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) ) ) → ( ( 𝑋 ∈ No ∧ 𝑌 ∈ No ) ∧ ( bday ‘ 𝑋 ) = ( bday ‘ 𝑌 ) ) ) |
64 |
|
nodense |
⊢ ( ( ( 𝑋 ∈ No ∧ 𝑌 ∈ No ) ∧ ( ( bday ‘ 𝑋 ) = ( bday ‘ 𝑌 ) ∧ 𝑋 <s 𝑌 ) ) → ∃ 𝑤 ∈ No ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) ) |
65 |
64
|
anassrs |
⊢ ( ( ( ( 𝑋 ∈ No ∧ 𝑌 ∈ No ) ∧ ( bday ‘ 𝑋 ) = ( bday ‘ 𝑌 ) ) ∧ 𝑋 <s 𝑌 ) → ∃ 𝑤 ∈ No ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) ) |
66 |
63 65
|
sylan |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ No ( ( 𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ∧ ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) ) ) ∧ 𝑋 <s 𝑌 ) → ∃ 𝑤 ∈ No ( ( bday ‘ 𝑤 ) ∈ ( bday ‘ 𝑋 ) ∧ 𝑋 <s 𝑤 ∧ 𝑤 <s 𝑌 ) ) |
67 |
55 66
|
mtand |
⊢ ( ( ( 𝐴 ⊆ No ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ No ( ( 𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) ) ∧ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ∧ ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) ) ) → ¬ 𝑋 <s 𝑌 ) |
68 |
67
|
ex |
⊢ ( ( 𝐴 ⊆ No ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ No ( ( 𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦 ) → 𝑧 ∈ 𝐴 ) ) → ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ∧ ( ( bday ‘ 𝑋 ) = ∩ ( bday “ 𝐴 ) ∧ ( bday ‘ 𝑌 ) = ∩ ( bday “ 𝐴 ) ) ) → ¬ 𝑋 <s 𝑌 ) ) |