Step |
Hyp |
Ref |
Expression |
1 |
|
simpl2 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ) → 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 ) = 1o ) ) → ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) |
4 |
|
3orel2 |
⊢ ( ¬ ( 𝐵 ‘ 𝑋 ) = 1o → ( ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) → ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) ) |
5 |
3 4
|
syl5com |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ) → ( ¬ ( 𝐵 ‘ 𝑋 ) = 1o → ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) ) |
6 |
|
simp13 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → 𝑋 ∈ On ) |
7 |
|
fveq1 |
⊢ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) → ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) = ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) ) |
8 |
7
|
adantr |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) = ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) ) |
9 |
|
simpr |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) → 𝑦 ∈ 𝑋 ) |
10 |
9
|
fvresd |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) ) |
11 |
9
|
fvresd |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) |
12 |
8 10 11
|
3eqtr3d |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) |
13 |
12
|
ralrimiva |
⊢ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) → ∀ 𝑦 ∈ 𝑋 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) |
14 |
13
|
adantr |
⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) → ∀ 𝑦 ∈ 𝑋 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) |
15 |
14
|
3ad2ant2 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ∀ 𝑦 ∈ 𝑋 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) |
16 |
|
simp2r |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( 𝐴 ‘ 𝑋 ) = 1o ) |
17 |
|
simp3 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) |
18 |
16 17
|
jca |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) ) |
19 |
|
andi |
⊢ ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) ↔ ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) ) |
20 |
18 19
|
sylib |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) ) |
21 |
|
3mix1 |
⊢ ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) → ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐴 ‘ 𝑋 ) = ∅ ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) ) |
22 |
|
3mix2 |
⊢ ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) → ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐴 ‘ 𝑋 ) = ∅ ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) ) |
23 |
21 22
|
jaoi |
⊢ ( ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐴 ‘ 𝑋 ) = ∅ ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) ) |
24 |
20 23
|
syl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐴 ‘ 𝑋 ) = ∅ ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) ) |
25 |
|
fvex |
⊢ ( 𝐴 ‘ 𝑋 ) ∈ V |
26 |
|
fvex |
⊢ ( 𝐵 ‘ 𝑋 ) ∈ V |
27 |
25 26
|
brtp |
⊢ ( ( 𝐴 ‘ 𝑋 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑋 ) ↔ ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐴 ‘ 𝑋 ) = ∅ ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) ) |
28 |
24 27
|
sylibr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( 𝐴 ‘ 𝑋 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑋 ) ) |
29 |
|
raleq |
⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) ) |
30 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝐴 ‘ 𝑥 ) = ( 𝐴 ‘ 𝑋 ) ) |
31 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝐵 ‘ 𝑥 ) = ( 𝐵 ‘ 𝑋 ) ) |
32 |
30 31
|
breq12d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ↔ ( 𝐴 ‘ 𝑋 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑋 ) ) ) |
33 |
29 32
|
anbi12d |
⊢ ( 𝑥 = 𝑋 → ( ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ 𝑋 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑋 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑋 ) ) ) ) |
34 |
33
|
rspcev |
⊢ ( ( 𝑋 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑋 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑋 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑋 ) ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ) |
35 |
6 15 28 34
|
syl12anc |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ) |
36 |
|
simp11 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → 𝐴 ∈ No ) |
37 |
|
simp12 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → 𝐵 ∈ No ) |
38 |
|
sltval |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ) ) |
39 |
36 37 38
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( 𝐴 <s 𝐵 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { 〈 1o , ∅ 〉 , 〈 1o , 2o 〉 , 〈 ∅ , 2o 〉 } ( 𝐵 ‘ 𝑥 ) ) ) ) |
40 |
35 39
|
mpbird |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → 𝐴 <s 𝐵 ) |
41 |
40
|
3expia |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ) → ( ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) → 𝐴 <s 𝐵 ) ) |
42 |
5 41
|
syld |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ) → ( ¬ ( 𝐵 ‘ 𝑋 ) = 1o → 𝐴 <s 𝐵 ) ) |
43 |
42
|
con1d |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ) → ( ¬ 𝐴 <s 𝐵 → ( 𝐵 ‘ 𝑋 ) = 1o ) ) |
44 |
43
|
3impia |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ¬ 𝐴 <s 𝐵 ) → ( 𝐵 ‘ 𝑋 ) = 1o ) |