Step |
Hyp |
Ref |
Expression |
1 |
|
simpl2 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) → 𝐵 ∈ No ) |
2 |
|
nofv |
⊢ ( 𝐵 ∈ No → ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) |
3 |
1 2
|
syl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) → ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) |
4 |
|
3orel3 |
⊢ ( ¬ ( 𝐵 ‘ 𝑋 ) = 2o → ( ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) → ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ) ) ) |
5 |
3 4
|
syl5com |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) → ( ¬ ( 𝐵 ‘ 𝑋 ) = 2o → ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ) ) ) |
6 |
|
simp13 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ) ) → 𝑋 ∈ On ) |
7 |
|
fveq1 |
⊢ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) → ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) = ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) ) |
8 |
7
|
eqcomd |
⊢ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) → ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) = ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) ) |
9 |
8
|
adantr |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) = ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) ) |
10 |
|
simpr |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) → 𝑦 ∈ 𝑋 ) |
11 |
10
|
fvresd |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) |
12 |
10
|
fvresd |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ) |
13 |
9 11 12
|
3eqtr3d |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝐵 ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ) |
14 |
13
|
ralrimiva |
⊢ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) → ∀ 𝑦 ∈ 𝑋 ( 𝐵 ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ) |
15 |
14
|
adantr |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) → ∀ 𝑦 ∈ 𝑋 ( 𝐵 ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ) |
16 |
15
|
3ad2ant2 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ) ) → ∀ 𝑦 ∈ 𝑋 ( 𝐵 ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ) |
17 |
|
simprr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) → ( 𝐴 ‘ 𝑋 ) = 2o ) |
18 |
17
|
a1d |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) → ( ( 𝐵 ‘ 𝑋 ) = ∅ → ( 𝐴 ‘ 𝑋 ) = 2o ) ) |
19 |
18
|
ancld |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) → ( ( 𝐵 ‘ 𝑋 ) = ∅ → ( ( 𝐵 ‘ 𝑋 ) = ∅ ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) ) |
20 |
17
|
a1d |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) → ( ( 𝐵 ‘ 𝑋 ) = 1o → ( 𝐴 ‘ 𝑋 ) = 2o ) ) |
21 |
20
|
ancld |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) → ( ( 𝐵 ‘ 𝑋 ) = 1o → ( ( 𝐵 ‘ 𝑋 ) = 1o ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) ) |
22 |
19 21
|
orim12d |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) → ( ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ) → ( ( ( 𝐵 ‘ 𝑋 ) = ∅ ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐵 ‘ 𝑋 ) = 1o ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) ) ) |
23 |
22
|
3impia |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ) ) → ( ( ( 𝐵 ‘ 𝑋 ) = ∅ ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐵 ‘ 𝑋 ) = 1o ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) ) |
24 |
|
3mix3 |
⊢ ( ( ( 𝐵 ‘ 𝑋 ) = ∅ ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) → ( ( ( 𝐵 ‘ 𝑋 ) = 1o ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐵 ‘ 𝑋 ) = 1o ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) ) |
25 |
|
3mix2 |
⊢ ( ( ( 𝐵 ‘ 𝑋 ) = 1o ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) → ( ( ( 𝐵 ‘ 𝑋 ) = 1o ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐵 ‘ 𝑋 ) = 1o ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) ) |
26 |
24 25
|
jaoi |
⊢ ( ( ( ( 𝐵 ‘ 𝑋 ) = ∅ ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐵 ‘ 𝑋 ) = 1o ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) → ( ( ( 𝐵 ‘ 𝑋 ) = 1o ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐵 ‘ 𝑋 ) = 1o ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) ) |
27 |
23 26
|
syl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ) ) → ( ( ( 𝐵 ‘ 𝑋 ) = 1o ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐵 ‘ 𝑋 ) = 1o ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) ) |
28 |
|
fvex |
⊢ ( 𝐵 ‘ 𝑋 ) ∈ V |
29 |
|
fvex |
⊢ ( 𝐴 ‘ 𝑋 ) ∈ V |
30 |
28 29
|
brtp |
⊢ ( ( 𝐵 ‘ 𝑋 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐴 ‘ 𝑋 ) ↔ ( ( ( 𝐵 ‘ 𝑋 ) = 1o ∧ ( 𝐴 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐵 ‘ 𝑋 ) = 1o ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) ) |
31 |
27 30
|
sylibr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ) ) → ( 𝐵 ‘ 𝑋 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐴 ‘ 𝑋 ) ) |
32 |
|
raleq |
⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑦 ∈ 𝑥 ( 𝐵 ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑋 ( 𝐵 ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ) ) |
33 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝐵 ‘ 𝑥 ) = ( 𝐵 ‘ 𝑋 ) ) |
34 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝐴 ‘ 𝑥 ) = ( 𝐴 ‘ 𝑋 ) ) |
35 |
33 34
|
breq12d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝐵 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐴 ‘ 𝑥 ) ↔ ( 𝐵 ‘ 𝑋 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐴 ‘ 𝑋 ) ) ) |
36 |
32 35
|
anbi12d |
⊢ ( 𝑥 = 𝑋 → ( ( ∀ 𝑦 ∈ 𝑥 ( 𝐵 ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ∧ ( 𝐵 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐴 ‘ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ 𝑋 ( 𝐵 ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ∧ ( 𝐵 ‘ 𝑋 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐴 ‘ 𝑋 ) ) ) ) |
37 |
36
|
rspcev |
⊢ ( ( 𝑋 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑋 ( 𝐵 ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ∧ ( 𝐵 ‘ 𝑋 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐴 ‘ 𝑋 ) ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐵 ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ∧ ( 𝐵 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐴 ‘ 𝑥 ) ) ) |
38 |
6 16 31 37
|
syl12anc |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐵 ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ∧ ( 𝐵 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐴 ‘ 𝑥 ) ) ) |
39 |
|
simp12 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ) ) → 𝐵 ∈ No ) |
40 |
|
simp11 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ) ) → 𝐴 ∈ No ) |
41 |
|
sltval |
⊢ ( ( 𝐵 ∈ No ∧ 𝐴 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐵 ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ∧ ( 𝐵 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐴 ‘ 𝑥 ) ) ) ) |
42 |
39 40 41
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ) ) → ( 𝐵 <s 𝐴 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐵 ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ∧ ( 𝐵 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐴 ‘ 𝑥 ) ) ) ) |
43 |
38 42
|
mpbird |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ) ) → 𝐵 <s 𝐴 ) |
44 |
43
|
3expia |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) → ( ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ) → 𝐵 <s 𝐴 ) ) |
45 |
5 44
|
syld |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) → ( ¬ ( 𝐵 ‘ 𝑋 ) = 2o → 𝐵 <s 𝐴 ) ) |
46 |
45
|
con1d |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ) → ( ¬ 𝐵 <s 𝐴 → ( 𝐵 ‘ 𝑋 ) = 2o ) ) |
47 |
46
|
3impia |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 2o ) ∧ ¬ 𝐵 <s 𝐴 ) → ( 𝐵 ‘ 𝑋 ) = 2o ) |