Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
⊢ ( 𝑥 = ∅ → ( 𝐴 +o 𝑥 ) = ( 𝐴 +o ∅ ) ) |
2 |
|
oveq2 |
⊢ ( 𝑥 = ∅ → ( 𝐵 +o 𝑥 ) = ( 𝐵 +o ∅ ) ) |
3 |
1 2
|
sseq12d |
⊢ ( 𝑥 = ∅ → ( ( 𝐴 +o 𝑥 ) ⊆ ( 𝐵 +o 𝑥 ) ↔ ( 𝐴 +o ∅ ) ⊆ ( 𝐵 +o ∅ ) ) ) |
4 |
3
|
imbi2d |
⊢ ( 𝑥 = ∅ → ( ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝑥 ) ⊆ ( 𝐵 +o 𝑥 ) ) ↔ ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o ∅ ) ⊆ ( 𝐵 +o ∅ ) ) ) ) |
5 |
4
|
imbi2d |
⊢ ( 𝑥 = ∅ → ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝑥 ) ⊆ ( 𝐵 +o 𝑥 ) ) ) ↔ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o ∅ ) ⊆ ( 𝐵 +o ∅ ) ) ) ) ) |
6 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐴 +o 𝑥 ) = ( 𝐴 +o 𝑦 ) ) |
7 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐵 +o 𝑥 ) = ( 𝐵 +o 𝑦 ) ) |
8 |
6 7
|
sseq12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐴 +o 𝑥 ) ⊆ ( 𝐵 +o 𝑥 ) ↔ ( 𝐴 +o 𝑦 ) ⊆ ( 𝐵 +o 𝑦 ) ) ) |
9 |
8
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝑥 ) ⊆ ( 𝐵 +o 𝑥 ) ) ↔ ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝑦 ) ⊆ ( 𝐵 +o 𝑦 ) ) ) ) |
10 |
9
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝑥 ) ⊆ ( 𝐵 +o 𝑥 ) ) ) ↔ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝑦 ) ⊆ ( 𝐵 +o 𝑦 ) ) ) ) ) |
11 |
|
oveq2 |
⊢ ( 𝑥 = suc 𝑦 → ( 𝐴 +o 𝑥 ) = ( 𝐴 +o suc 𝑦 ) ) |
12 |
|
oveq2 |
⊢ ( 𝑥 = suc 𝑦 → ( 𝐵 +o 𝑥 ) = ( 𝐵 +o suc 𝑦 ) ) |
13 |
11 12
|
sseq12d |
⊢ ( 𝑥 = suc 𝑦 → ( ( 𝐴 +o 𝑥 ) ⊆ ( 𝐵 +o 𝑥 ) ↔ ( 𝐴 +o suc 𝑦 ) ⊆ ( 𝐵 +o suc 𝑦 ) ) ) |
14 |
13
|
imbi2d |
⊢ ( 𝑥 = suc 𝑦 → ( ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝑥 ) ⊆ ( 𝐵 +o 𝑥 ) ) ↔ ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o suc 𝑦 ) ⊆ ( 𝐵 +o suc 𝑦 ) ) ) ) |
15 |
14
|
imbi2d |
⊢ ( 𝑥 = suc 𝑦 → ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝑥 ) ⊆ ( 𝐵 +o 𝑥 ) ) ) ↔ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o suc 𝑦 ) ⊆ ( 𝐵 +o suc 𝑦 ) ) ) ) ) |
16 |
|
oveq2 |
⊢ ( 𝑥 = 𝐶 → ( 𝐴 +o 𝑥 ) = ( 𝐴 +o 𝐶 ) ) |
17 |
|
oveq2 |
⊢ ( 𝑥 = 𝐶 → ( 𝐵 +o 𝑥 ) = ( 𝐵 +o 𝐶 ) ) |
18 |
16 17
|
sseq12d |
⊢ ( 𝑥 = 𝐶 → ( ( 𝐴 +o 𝑥 ) ⊆ ( 𝐵 +o 𝑥 ) ↔ ( 𝐴 +o 𝐶 ) ⊆ ( 𝐵 +o 𝐶 ) ) ) |
19 |
18
|
imbi2d |
⊢ ( 𝑥 = 𝐶 → ( ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝑥 ) ⊆ ( 𝐵 +o 𝑥 ) ) ↔ ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝐶 ) ⊆ ( 𝐵 +o 𝐶 ) ) ) ) |
20 |
19
|
imbi2d |
⊢ ( 𝑥 = 𝐶 → ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝑥 ) ⊆ ( 𝐵 +o 𝑥 ) ) ) ↔ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝐶 ) ⊆ ( 𝐵 +o 𝐶 ) ) ) ) ) |
21 |
|
nnon |
⊢ ( 𝐴 ∈ ω → 𝐴 ∈ On ) |
22 |
|
nnon |
⊢ ( 𝐵 ∈ ω → 𝐵 ∈ On ) |
23 |
|
oa0 |
⊢ ( 𝐴 ∈ On → ( 𝐴 +o ∅ ) = 𝐴 ) |
24 |
23
|
adantr |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +o ∅ ) = 𝐴 ) |
25 |
|
oa0 |
⊢ ( 𝐵 ∈ On → ( 𝐵 +o ∅ ) = 𝐵 ) |
26 |
25
|
adantl |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 +o ∅ ) = 𝐵 ) |
27 |
24 26
|
sseq12d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +o ∅ ) ⊆ ( 𝐵 +o ∅ ) ↔ 𝐴 ⊆ 𝐵 ) ) |
28 |
27
|
biimprd |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o ∅ ) ⊆ ( 𝐵 +o ∅ ) ) ) |
29 |
21 22 28
|
syl2an |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o ∅ ) ⊆ ( 𝐵 +o ∅ ) ) ) |
30 |
|
nnacl |
⊢ ( ( 𝐴 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝐴 +o 𝑦 ) ∈ ω ) |
31 |
30
|
ancoms |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐴 ∈ ω ) → ( 𝐴 +o 𝑦 ) ∈ ω ) |
32 |
31
|
adantrr |
⊢ ( ( 𝑦 ∈ ω ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝐴 +o 𝑦 ) ∈ ω ) |
33 |
|
nnon |
⊢ ( ( 𝐴 +o 𝑦 ) ∈ ω → ( 𝐴 +o 𝑦 ) ∈ On ) |
34 |
|
eloni |
⊢ ( ( 𝐴 +o 𝑦 ) ∈ On → Ord ( 𝐴 +o 𝑦 ) ) |
35 |
32 33 34
|
3syl |
⊢ ( ( 𝑦 ∈ ω ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → Ord ( 𝐴 +o 𝑦 ) ) |
36 |
|
nnacl |
⊢ ( ( 𝐵 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝐵 +o 𝑦 ) ∈ ω ) |
37 |
36
|
ancoms |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐵 +o 𝑦 ) ∈ ω ) |
38 |
37
|
adantrl |
⊢ ( ( 𝑦 ∈ ω ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝐵 +o 𝑦 ) ∈ ω ) |
39 |
|
nnon |
⊢ ( ( 𝐵 +o 𝑦 ) ∈ ω → ( 𝐵 +o 𝑦 ) ∈ On ) |
40 |
|
eloni |
⊢ ( ( 𝐵 +o 𝑦 ) ∈ On → Ord ( 𝐵 +o 𝑦 ) ) |
41 |
38 39 40
|
3syl |
⊢ ( ( 𝑦 ∈ ω ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → Ord ( 𝐵 +o 𝑦 ) ) |
42 |
|
ordsucsssuc |
⊢ ( ( Ord ( 𝐴 +o 𝑦 ) ∧ Ord ( 𝐵 +o 𝑦 ) ) → ( ( 𝐴 +o 𝑦 ) ⊆ ( 𝐵 +o 𝑦 ) ↔ suc ( 𝐴 +o 𝑦 ) ⊆ suc ( 𝐵 +o 𝑦 ) ) ) |
43 |
35 41 42
|
syl2anc |
⊢ ( ( 𝑦 ∈ ω ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝐴 +o 𝑦 ) ⊆ ( 𝐵 +o 𝑦 ) ↔ suc ( 𝐴 +o 𝑦 ) ⊆ suc ( 𝐵 +o 𝑦 ) ) ) |
44 |
43
|
biimpa |
⊢ ( ( ( 𝑦 ∈ ω ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) ∧ ( 𝐴 +o 𝑦 ) ⊆ ( 𝐵 +o 𝑦 ) ) → suc ( 𝐴 +o 𝑦 ) ⊆ suc ( 𝐵 +o 𝑦 ) ) |
45 |
|
nnasuc |
⊢ ( ( 𝐴 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝐴 +o suc 𝑦 ) = suc ( 𝐴 +o 𝑦 ) ) |
46 |
45
|
ancoms |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐴 ∈ ω ) → ( 𝐴 +o suc 𝑦 ) = suc ( 𝐴 +o 𝑦 ) ) |
47 |
46
|
adantrr |
⊢ ( ( 𝑦 ∈ ω ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝐴 +o suc 𝑦 ) = suc ( 𝐴 +o 𝑦 ) ) |
48 |
|
nnasuc |
⊢ ( ( 𝐵 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝐵 +o suc 𝑦 ) = suc ( 𝐵 +o 𝑦 ) ) |
49 |
48
|
ancoms |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐵 +o suc 𝑦 ) = suc ( 𝐵 +o 𝑦 ) ) |
50 |
49
|
adantrl |
⊢ ( ( 𝑦 ∈ ω ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝐵 +o suc 𝑦 ) = suc ( 𝐵 +o 𝑦 ) ) |
51 |
47 50
|
sseq12d |
⊢ ( ( 𝑦 ∈ ω ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝐴 +o suc 𝑦 ) ⊆ ( 𝐵 +o suc 𝑦 ) ↔ suc ( 𝐴 +o 𝑦 ) ⊆ suc ( 𝐵 +o 𝑦 ) ) ) |
52 |
51
|
adantr |
⊢ ( ( ( 𝑦 ∈ ω ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) ∧ ( 𝐴 +o 𝑦 ) ⊆ ( 𝐵 +o 𝑦 ) ) → ( ( 𝐴 +o suc 𝑦 ) ⊆ ( 𝐵 +o suc 𝑦 ) ↔ suc ( 𝐴 +o 𝑦 ) ⊆ suc ( 𝐵 +o 𝑦 ) ) ) |
53 |
44 52
|
mpbird |
⊢ ( ( ( 𝑦 ∈ ω ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) ∧ ( 𝐴 +o 𝑦 ) ⊆ ( 𝐵 +o 𝑦 ) ) → ( 𝐴 +o suc 𝑦 ) ⊆ ( 𝐵 +o suc 𝑦 ) ) |
54 |
53
|
ex |
⊢ ( ( 𝑦 ∈ ω ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝐴 +o 𝑦 ) ⊆ ( 𝐵 +o 𝑦 ) → ( 𝐴 +o suc 𝑦 ) ⊆ ( 𝐵 +o suc 𝑦 ) ) ) |
55 |
54
|
imim2d |
⊢ ( ( 𝑦 ∈ ω ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝑦 ) ⊆ ( 𝐵 +o 𝑦 ) ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o suc 𝑦 ) ⊆ ( 𝐵 +o suc 𝑦 ) ) ) ) |
56 |
55
|
ex |
⊢ ( 𝑦 ∈ ω → ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝑦 ) ⊆ ( 𝐵 +o 𝑦 ) ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o suc 𝑦 ) ⊆ ( 𝐵 +o suc 𝑦 ) ) ) ) ) |
57 |
56
|
a2d |
⊢ ( 𝑦 ∈ ω → ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝑦 ) ⊆ ( 𝐵 +o 𝑦 ) ) ) → ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o suc 𝑦 ) ⊆ ( 𝐵 +o suc 𝑦 ) ) ) ) ) |
58 |
5 10 15 20 29 57
|
finds |
⊢ ( 𝐶 ∈ ω → ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝐶 ) ⊆ ( 𝐵 +o 𝐶 ) ) ) ) |
59 |
58
|
com12 |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐶 ∈ ω → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝐶 ) ⊆ ( 𝐵 +o 𝐶 ) ) ) ) |
60 |
59
|
3impia |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +o 𝐶 ) ⊆ ( 𝐵 +o 𝐶 ) ) ) |