Step |
Hyp |
Ref |
Expression |
1 |
|
nodenselem6 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ No ) |
2 |
|
bdayval |
⊢ ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ No → ( bday ‘ ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) = dom ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) |
3 |
1 2
|
syl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( bday ‘ ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) = dom ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) |
4 |
|
dmres |
⊢ dom ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∩ dom 𝐴 ) |
5 |
|
nodenselem5 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) ) |
6 |
|
bdayfo |
⊢ bday : No –onto→ On |
7 |
|
fof |
⊢ ( bday : No –onto→ On → bday : No ⟶ On ) |
8 |
6 7
|
ax-mp |
⊢ bday : No ⟶ On |
9 |
|
0elon |
⊢ ∅ ∈ On |
10 |
8 9
|
f0cli |
⊢ ( bday ‘ 𝐴 ) ∈ On |
11 |
10
|
onelssi |
⊢ ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ ( bday ‘ 𝐴 ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ⊆ ( bday ‘ 𝐴 ) ) |
12 |
5 11
|
syl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ⊆ ( bday ‘ 𝐴 ) ) |
13 |
|
bdayval |
⊢ ( 𝐴 ∈ No → ( bday ‘ 𝐴 ) = dom 𝐴 ) |
14 |
13
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( bday ‘ 𝐴 ) = dom 𝐴 ) |
15 |
12 14
|
sseqtrd |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ⊆ dom 𝐴 ) |
16 |
|
df-ss |
⊢ ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ⊆ dom 𝐴 ↔ ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∩ dom 𝐴 ) = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) |
17 |
15 16
|
sylib |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∩ dom 𝐴 ) = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) |
18 |
4 17
|
syl5eq |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → dom ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) |
19 |
3 18
|
eqtrd |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( bday ‘ ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) |
20 |
19 5
|
eqeltrd |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( bday ‘ ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ∈ ( bday ‘ 𝐴 ) ) |
21 |
|
nodenselem4 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵 ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On ) |
22 |
21
|
adantrl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On ) |
23 |
|
nodenselem8 |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ) → ( 𝐴 <s 𝐵 ↔ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) ) ) |
24 |
23
|
biimpd |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ) → ( 𝐴 <s 𝐵 → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) ) ) |
25 |
24
|
3expia |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) → ( 𝐴 <s 𝐵 → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) ) ) ) |
26 |
25
|
imp32 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) ) |
27 |
26
|
simpld |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ) |
28 |
|
eqid |
⊢ ∅ = ∅ |
29 |
27 28
|
jctir |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ∧ ∅ = ∅ ) ) |
30 |
29
|
3mix1d |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ∧ ∅ = ∅ ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ∧ ∅ = 2o ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ∧ ∅ = 2o ) ) ) |
31 |
|
fvex |
⊢ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ V |
32 |
|
0ex |
⊢ ∅ ∈ V |
33 |
31 32
|
brtp |
⊢ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ∅ ↔ ( ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ∧ ∅ = ∅ ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 1o ∧ ∅ = 2o ) ∨ ( ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ∧ ∅ = 2o ) ) ) |
34 |
30 33
|
sylibr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ∅ ) |
35 |
19
|
fveq2d |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ ( bday ‘ ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) = ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) |
36 |
|
fvnobday |
⊢ ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ No → ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ ( bday ‘ ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) = ∅ ) |
37 |
1 36
|
syl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ ( bday ‘ ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) = ∅ ) |
38 |
35 37
|
eqtr3d |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) |
39 |
34 38
|
breqtrrd |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) |
40 |
|
fvres |
⊢ ( 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ) |
41 |
40
|
eqcomd |
⊢ ( 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( 𝐴 ‘ 𝑦 ) = ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) ) |
42 |
41
|
rgen |
⊢ ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) |
43 |
39 42
|
jctil |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) |
44 |
|
raleq |
⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) ) ) |
45 |
|
fveq2 |
⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( 𝐴 ‘ 𝑥 ) = ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) |
46 |
|
fveq2 |
⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑥 ) = ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) |
47 |
45 46
|
breq12d |
⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑥 ) ↔ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) |
48 |
44 47
|
anbi12d |
⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) ) |
49 |
48
|
rspcev |
⊢ ( ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On ∧ ( ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( 𝐴 ‘ 𝑦 ) = ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) ∧ ( 𝐴 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑥 ) ) ) |
50 |
22 43 49
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑥 ) ) ) |
51 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → 𝐴 ∈ No ) |
52 |
|
sltval |
⊢ ( ( 𝐴 ∈ No ∧ ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ No ) → ( 𝐴 <s ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑥 ) ) ) ) |
53 |
51 1 52
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( 𝐴 <s ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑥 ) ) ) ) |
54 |
50 53
|
mpbird |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → 𝐴 <s ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) |
55 |
41
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) ∧ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ( 𝐴 ‘ 𝑦 ) = ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) ) |
56 |
|
nodenselem7 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ) |
57 |
56
|
imp |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) ∧ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) |
58 |
55 57
|
eqtr3d |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) ∧ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) |
59 |
58
|
ralrimiva |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) |
60 |
26
|
simprd |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) |
61 |
60 28
|
jctil |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( ∅ = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) ) |
62 |
61
|
3mix3d |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( ( ∅ = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) ∨ ( ∅ = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) ∨ ( ∅ = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) ) ) |
63 |
|
fvex |
⊢ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ V |
64 |
32 63
|
brtp |
⊢ ( ∅ { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ↔ ( ( ∅ = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = ∅ ) ∨ ( ∅ = 1o ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) ∨ ( ∅ = ∅ ∧ ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) = 2o ) ) ) |
65 |
62 64
|
sylibr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ∅ { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) |
66 |
38 65
|
eqbrtrd |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) |
67 |
|
raleq |
⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( ∀ 𝑦 ∈ 𝑥 ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ) |
68 |
|
fveq2 |
⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( 𝐵 ‘ 𝑥 ) = ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) |
69 |
46 68
|
breq12d |
⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ↔ ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) |
70 |
67 69
|
anbi12d |
⊢ ( 𝑥 = ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } → ( ( ∀ 𝑦 ∈ 𝑥 ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) ) |
71 |
70
|
rspcev |
⊢ ( ( ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ∈ On ∧ ( ∀ 𝑦 ∈ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ) |
72 |
22 59 66 71
|
syl12anc |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ) |
73 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → 𝐵 ∈ No ) |
74 |
|
sltval |
⊢ ( ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ No ∧ 𝐵 ∈ No ) → ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) <s 𝐵 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ) ) |
75 |
1 73 74
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) <s 𝐵 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ) ) |
76 |
72 75
|
mpbird |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) <s 𝐵 ) |
77 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ( bday ‘ 𝑥 ) = ( bday ‘ ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) |
78 |
77
|
eleq1d |
⊢ ( 𝑥 = ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ( ( bday ‘ 𝑥 ) ∈ ( bday ‘ 𝐴 ) ↔ ( bday ‘ ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ∈ ( bday ‘ 𝐴 ) ) ) |
79 |
|
breq2 |
⊢ ( 𝑥 = ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ( 𝐴 <s 𝑥 ↔ 𝐴 <s ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ) |
80 |
|
breq1 |
⊢ ( 𝑥 = ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ( 𝑥 <s 𝐵 ↔ ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) <s 𝐵 ) ) |
81 |
78 79 80
|
3anbi123d |
⊢ ( 𝑥 = ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) → ( ( ( bday ‘ 𝑥 ) ∈ ( bday ‘ 𝐴 ) ∧ 𝐴 <s 𝑥 ∧ 𝑥 <s 𝐵 ) ↔ ( ( bday ‘ ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ∈ ( bday ‘ 𝐴 ) ∧ 𝐴 <s ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∧ ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) <s 𝐵 ) ) ) |
82 |
81
|
rspcev |
⊢ ( ( ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∈ No ∧ ( ( bday ‘ ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ) ∈ ( bday ‘ 𝐴 ) ∧ 𝐴 <s ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) ∧ ( 𝐴 ↾ ∩ { 𝑎 ∈ On ∣ ( 𝐴 ‘ 𝑎 ) ≠ ( 𝐵 ‘ 𝑎 ) } ) <s 𝐵 ) ) → ∃ 𝑥 ∈ No ( ( bday ‘ 𝑥 ) ∈ ( bday ‘ 𝐴 ) ∧ 𝐴 <s 𝑥 ∧ 𝑥 <s 𝐵 ) ) |
83 |
1 20 54 76 82
|
syl13anc |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( bday ‘ 𝐴 ) = ( bday ‘ 𝐵 ) ∧ 𝐴 <s 𝐵 ) ) → ∃ 𝑥 ∈ No ( ( bday ‘ 𝑥 ) ∈ ( bday ‘ 𝐴 ) ∧ 𝐴 <s 𝑥 ∧ 𝑥 <s 𝐵 ) ) |