Step |
Hyp |
Ref |
Expression |
1 |
|
findsg.1 |
⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜓 ) ) |
2 |
|
findsg.2 |
⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜒 ) ) |
3 |
|
findsg.3 |
⊢ ( 𝑥 = suc 𝑦 → ( 𝜑 ↔ 𝜃 ) ) |
4 |
|
findsg.4 |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜏 ) ) |
5 |
|
findsg.5 |
⊢ ( 𝐵 ∈ ω → 𝜓 ) |
6 |
|
findsg.6 |
⊢ ( ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝐵 ⊆ 𝑦 ) → ( 𝜒 → 𝜃 ) ) |
7 |
|
sseq2 |
⊢ ( 𝑥 = ∅ → ( 𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ ∅ ) ) |
8 |
7
|
adantl |
⊢ ( ( 𝐵 = ∅ ∧ 𝑥 = ∅ ) → ( 𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ ∅ ) ) |
9 |
|
eqeq2 |
⊢ ( 𝐵 = ∅ → ( 𝑥 = 𝐵 ↔ 𝑥 = ∅ ) ) |
10 |
9 1
|
syl6bir |
⊢ ( 𝐵 = ∅ → ( 𝑥 = ∅ → ( 𝜑 ↔ 𝜓 ) ) ) |
11 |
10
|
imp |
⊢ ( ( 𝐵 = ∅ ∧ 𝑥 = ∅ ) → ( 𝜑 ↔ 𝜓 ) ) |
12 |
8 11
|
imbi12d |
⊢ ( ( 𝐵 = ∅ ∧ 𝑥 = ∅ ) → ( ( 𝐵 ⊆ 𝑥 → 𝜑 ) ↔ ( 𝐵 ⊆ ∅ → 𝜓 ) ) ) |
13 |
7
|
imbi1d |
⊢ ( 𝑥 = ∅ → ( ( 𝐵 ⊆ 𝑥 → 𝜑 ) ↔ ( 𝐵 ⊆ ∅ → 𝜑 ) ) ) |
14 |
|
ss0 |
⊢ ( 𝐵 ⊆ ∅ → 𝐵 = ∅ ) |
15 |
14
|
con3i |
⊢ ( ¬ 𝐵 = ∅ → ¬ 𝐵 ⊆ ∅ ) |
16 |
15
|
pm2.21d |
⊢ ( ¬ 𝐵 = ∅ → ( 𝐵 ⊆ ∅ → ( 𝜑 ↔ 𝜓 ) ) ) |
17 |
16
|
pm5.74d |
⊢ ( ¬ 𝐵 = ∅ → ( ( 𝐵 ⊆ ∅ → 𝜑 ) ↔ ( 𝐵 ⊆ ∅ → 𝜓 ) ) ) |
18 |
13 17
|
sylan9bbr |
⊢ ( ( ¬ 𝐵 = ∅ ∧ 𝑥 = ∅ ) → ( ( 𝐵 ⊆ 𝑥 → 𝜑 ) ↔ ( 𝐵 ⊆ ∅ → 𝜓 ) ) ) |
19 |
12 18
|
pm2.61ian |
⊢ ( 𝑥 = ∅ → ( ( 𝐵 ⊆ 𝑥 → 𝜑 ) ↔ ( 𝐵 ⊆ ∅ → 𝜓 ) ) ) |
20 |
19
|
imbi2d |
⊢ ( 𝑥 = ∅ → ( ( 𝐵 ∈ ω → ( 𝐵 ⊆ 𝑥 → 𝜑 ) ) ↔ ( 𝐵 ∈ ω → ( 𝐵 ⊆ ∅ → 𝜓 ) ) ) ) |
21 |
|
sseq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝑦 ) ) |
22 |
21 2
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐵 ⊆ 𝑥 → 𝜑 ) ↔ ( 𝐵 ⊆ 𝑦 → 𝜒 ) ) ) |
23 |
22
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐵 ∈ ω → ( 𝐵 ⊆ 𝑥 → 𝜑 ) ) ↔ ( 𝐵 ∈ ω → ( 𝐵 ⊆ 𝑦 → 𝜒 ) ) ) ) |
24 |
|
sseq2 |
⊢ ( 𝑥 = suc 𝑦 → ( 𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ suc 𝑦 ) ) |
25 |
24 3
|
imbi12d |
⊢ ( 𝑥 = suc 𝑦 → ( ( 𝐵 ⊆ 𝑥 → 𝜑 ) ↔ ( 𝐵 ⊆ suc 𝑦 → 𝜃 ) ) ) |
26 |
25
|
imbi2d |
⊢ ( 𝑥 = suc 𝑦 → ( ( 𝐵 ∈ ω → ( 𝐵 ⊆ 𝑥 → 𝜑 ) ) ↔ ( 𝐵 ∈ ω → ( 𝐵 ⊆ suc 𝑦 → 𝜃 ) ) ) ) |
27 |
|
sseq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝐴 ) ) |
28 |
27 4
|
imbi12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝐵 ⊆ 𝑥 → 𝜑 ) ↔ ( 𝐵 ⊆ 𝐴 → 𝜏 ) ) ) |
29 |
28
|
imbi2d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝐵 ∈ ω → ( 𝐵 ⊆ 𝑥 → 𝜑 ) ) ↔ ( 𝐵 ∈ ω → ( 𝐵 ⊆ 𝐴 → 𝜏 ) ) ) ) |
30 |
5
|
a1d |
⊢ ( 𝐵 ∈ ω → ( 𝐵 ⊆ ∅ → 𝜓 ) ) |
31 |
|
vex |
⊢ 𝑦 ∈ V |
32 |
31
|
sucex |
⊢ suc 𝑦 ∈ V |
33 |
32
|
eqvinc |
⊢ ( suc 𝑦 = 𝐵 ↔ ∃ 𝑥 ( 𝑥 = suc 𝑦 ∧ 𝑥 = 𝐵 ) ) |
34 |
5 1
|
syl5ibr |
⊢ ( 𝑥 = 𝐵 → ( 𝐵 ∈ ω → 𝜑 ) ) |
35 |
3
|
biimpd |
⊢ ( 𝑥 = suc 𝑦 → ( 𝜑 → 𝜃 ) ) |
36 |
34 35
|
sylan9r |
⊢ ( ( 𝑥 = suc 𝑦 ∧ 𝑥 = 𝐵 ) → ( 𝐵 ∈ ω → 𝜃 ) ) |
37 |
36
|
exlimiv |
⊢ ( ∃ 𝑥 ( 𝑥 = suc 𝑦 ∧ 𝑥 = 𝐵 ) → ( 𝐵 ∈ ω → 𝜃 ) ) |
38 |
33 37
|
sylbi |
⊢ ( suc 𝑦 = 𝐵 → ( 𝐵 ∈ ω → 𝜃 ) ) |
39 |
38
|
eqcoms |
⊢ ( 𝐵 = suc 𝑦 → ( 𝐵 ∈ ω → 𝜃 ) ) |
40 |
39
|
imim2i |
⊢ ( ( 𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦 ) → ( 𝐵 ⊆ suc 𝑦 → ( 𝐵 ∈ ω → 𝜃 ) ) ) |
41 |
40
|
a1d |
⊢ ( ( 𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦 ) → ( ( 𝐵 ⊆ 𝑦 → 𝜒 ) → ( 𝐵 ⊆ suc 𝑦 → ( 𝐵 ∈ ω → 𝜃 ) ) ) ) |
42 |
41
|
com4r |
⊢ ( 𝐵 ∈ ω → ( ( 𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦 ) → ( ( 𝐵 ⊆ 𝑦 → 𝜒 ) → ( 𝐵 ⊆ suc 𝑦 → 𝜃 ) ) ) ) |
43 |
42
|
adantl |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦 ) → ( ( 𝐵 ⊆ 𝑦 → 𝜒 ) → ( 𝐵 ⊆ suc 𝑦 → 𝜃 ) ) ) ) |
44 |
|
df-ne |
⊢ ( 𝐵 ≠ suc 𝑦 ↔ ¬ 𝐵 = suc 𝑦 ) |
45 |
44
|
anbi2i |
⊢ ( ( 𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦 ) ↔ ( 𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦 ) ) |
46 |
|
annim |
⊢ ( ( 𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦 ) ↔ ¬ ( 𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦 ) ) |
47 |
45 46
|
bitri |
⊢ ( ( 𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦 ) ↔ ¬ ( 𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦 ) ) |
48 |
|
nnon |
⊢ ( 𝐵 ∈ ω → 𝐵 ∈ On ) |
49 |
|
nnon |
⊢ ( 𝑦 ∈ ω → 𝑦 ∈ On ) |
50 |
|
onsssuc |
⊢ ( ( 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐵 ⊆ 𝑦 ↔ 𝐵 ∈ suc 𝑦 ) ) |
51 |
|
suceloni |
⊢ ( 𝑦 ∈ On → suc 𝑦 ∈ On ) |
52 |
|
onelpss |
⊢ ( ( 𝐵 ∈ On ∧ suc 𝑦 ∈ On ) → ( 𝐵 ∈ suc 𝑦 ↔ ( 𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦 ) ) ) |
53 |
51 52
|
sylan2 |
⊢ ( ( 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐵 ∈ suc 𝑦 ↔ ( 𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦 ) ) ) |
54 |
50 53
|
bitrd |
⊢ ( ( 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐵 ⊆ 𝑦 ↔ ( 𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦 ) ) ) |
55 |
48 49 54
|
syl2anr |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐵 ⊆ 𝑦 ↔ ( 𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦 ) ) ) |
56 |
6
|
ex |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐵 ⊆ 𝑦 → ( 𝜒 → 𝜃 ) ) ) |
57 |
56
|
a1ddd |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐵 ⊆ 𝑦 → ( 𝜒 → ( 𝐵 ⊆ suc 𝑦 → 𝜃 ) ) ) ) |
58 |
57
|
a2d |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐵 ⊆ 𝑦 → 𝜒 ) → ( 𝐵 ⊆ 𝑦 → ( 𝐵 ⊆ suc 𝑦 → 𝜃 ) ) ) ) |
59 |
58
|
com23 |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐵 ⊆ 𝑦 → ( ( 𝐵 ⊆ 𝑦 → 𝜒 ) → ( 𝐵 ⊆ suc 𝑦 → 𝜃 ) ) ) ) |
60 |
55 59
|
sylbird |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦 ) → ( ( 𝐵 ⊆ 𝑦 → 𝜒 ) → ( 𝐵 ⊆ suc 𝑦 → 𝜃 ) ) ) ) |
61 |
47 60
|
syl5bir |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) → ( ¬ ( 𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦 ) → ( ( 𝐵 ⊆ 𝑦 → 𝜒 ) → ( 𝐵 ⊆ suc 𝑦 → 𝜃 ) ) ) ) |
62 |
43 61
|
pm2.61d |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐵 ⊆ 𝑦 → 𝜒 ) → ( 𝐵 ⊆ suc 𝑦 → 𝜃 ) ) ) |
63 |
62
|
ex |
⊢ ( 𝑦 ∈ ω → ( 𝐵 ∈ ω → ( ( 𝐵 ⊆ 𝑦 → 𝜒 ) → ( 𝐵 ⊆ suc 𝑦 → 𝜃 ) ) ) ) |
64 |
63
|
a2d |
⊢ ( 𝑦 ∈ ω → ( ( 𝐵 ∈ ω → ( 𝐵 ⊆ 𝑦 → 𝜒 ) ) → ( 𝐵 ∈ ω → ( 𝐵 ⊆ suc 𝑦 → 𝜃 ) ) ) ) |
65 |
20 23 26 29 30 64
|
finds |
⊢ ( 𝐴 ∈ ω → ( 𝐵 ∈ ω → ( 𝐵 ⊆ 𝐴 → 𝜏 ) ) ) |
66 |
65
|
imp31 |
⊢ ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝐵 ⊆ 𝐴 ) → 𝜏 ) |