Step |
Hyp |
Ref |
Expression |
1 |
|
zorn2lem.3 |
⊢ 𝐹 = recs ( ( 𝑓 ∈ V ↦ ( ℩ 𝑣 ∈ 𝐶 ∀ 𝑢 ∈ 𝐶 ¬ 𝑢 𝑤 𝑣 ) ) ) |
2 |
|
zorn2lem.4 |
⊢ 𝐶 = { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ran 𝑓 𝑔 𝑅 𝑧 } |
3 |
|
zorn2lem.5 |
⊢ 𝐷 = { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑥 ) 𝑔 𝑅 𝑧 } |
4 |
|
zorn2lem.7 |
⊢ 𝐻 = { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑦 ) 𝑔 𝑅 𝑧 } |
5 |
|
poss |
⊢ ( ( 𝐹 “ 𝑥 ) ⊆ 𝐴 → ( 𝑅 Po 𝐴 → 𝑅 Po ( 𝐹 “ 𝑥 ) ) ) |
6 |
1 2 3 4
|
zorn2lem5 |
⊢ ( ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) ∧ ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ ) → ( 𝐹 “ 𝑥 ) ⊆ 𝐴 ) |
7 |
5 6
|
syl11 |
⊢ ( 𝑅 Po 𝐴 → ( ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) ∧ ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ ) → 𝑅 Po ( 𝐹 “ 𝑥 ) ) ) |
8 |
1
|
tfr1 |
⊢ 𝐹 Fn On |
9 |
|
fnfun |
⊢ ( 𝐹 Fn On → Fun 𝐹 ) |
10 |
|
fvelima |
⊢ ( ( Fun 𝐹 ∧ 𝑠 ∈ ( 𝐹 “ 𝑥 ) ) → ∃ 𝑏 ∈ 𝑥 ( 𝐹 ‘ 𝑏 ) = 𝑠 ) |
11 |
|
df-rex |
⊢ ( ∃ 𝑏 ∈ 𝑥 ( 𝐹 ‘ 𝑏 ) = 𝑠 ↔ ∃ 𝑏 ( 𝑏 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑠 ) ) |
12 |
10 11
|
sylib |
⊢ ( ( Fun 𝐹 ∧ 𝑠 ∈ ( 𝐹 “ 𝑥 ) ) → ∃ 𝑏 ( 𝑏 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑠 ) ) |
13 |
12
|
ex |
⊢ ( Fun 𝐹 → ( 𝑠 ∈ ( 𝐹 “ 𝑥 ) → ∃ 𝑏 ( 𝑏 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑠 ) ) ) |
14 |
|
fvelima |
⊢ ( ( Fun 𝐹 ∧ 𝑟 ∈ ( 𝐹 “ 𝑥 ) ) → ∃ 𝑎 ∈ 𝑥 ( 𝐹 ‘ 𝑎 ) = 𝑟 ) |
15 |
|
df-rex |
⊢ ( ∃ 𝑎 ∈ 𝑥 ( 𝐹 ‘ 𝑎 ) = 𝑟 ↔ ∃ 𝑎 ( 𝑎 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) |
16 |
14 15
|
sylib |
⊢ ( ( Fun 𝐹 ∧ 𝑟 ∈ ( 𝐹 “ 𝑥 ) ) → ∃ 𝑎 ( 𝑎 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) |
17 |
16
|
ex |
⊢ ( Fun 𝐹 → ( 𝑟 ∈ ( 𝐹 “ 𝑥 ) → ∃ 𝑎 ( 𝑎 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) ) |
18 |
13 17
|
anim12d |
⊢ ( Fun 𝐹 → ( ( 𝑠 ∈ ( 𝐹 “ 𝑥 ) ∧ 𝑟 ∈ ( 𝐹 “ 𝑥 ) ) → ( ∃ 𝑏 ( 𝑏 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑠 ) ∧ ∃ 𝑎 ( 𝑎 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) ) ) |
19 |
8 9 18
|
mp2b |
⊢ ( ( 𝑠 ∈ ( 𝐹 “ 𝑥 ) ∧ 𝑟 ∈ ( 𝐹 “ 𝑥 ) ) → ( ∃ 𝑏 ( 𝑏 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑠 ) ∧ ∃ 𝑎 ( 𝑎 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) ) |
20 |
|
an4 |
⊢ ( ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) ∧ ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) ↔ ( ( 𝑏 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑠 ) ∧ ( 𝑎 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) ) |
21 |
20
|
2exbii |
⊢ ( ∃ 𝑏 ∃ 𝑎 ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) ∧ ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) ↔ ∃ 𝑏 ∃ 𝑎 ( ( 𝑏 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑠 ) ∧ ( 𝑎 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) ) |
22 |
|
exdistrv |
⊢ ( ∃ 𝑏 ∃ 𝑎 ( ( 𝑏 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑠 ) ∧ ( 𝑎 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) ↔ ( ∃ 𝑏 ( 𝑏 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑠 ) ∧ ∃ 𝑎 ( 𝑎 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) ) |
23 |
21 22
|
bitri |
⊢ ( ∃ 𝑏 ∃ 𝑎 ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) ∧ ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) ↔ ( ∃ 𝑏 ( 𝑏 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑠 ) ∧ ∃ 𝑎 ( 𝑎 ∈ 𝑥 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) ) |
24 |
19 23
|
sylibr |
⊢ ( ( 𝑠 ∈ ( 𝐹 “ 𝑥 ) ∧ 𝑟 ∈ ( 𝐹 “ 𝑥 ) ) → ∃ 𝑏 ∃ 𝑎 ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) ∧ ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) ) |
25 |
4
|
neeq1i |
⊢ ( 𝐻 ≠ ∅ ↔ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑦 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) |
26 |
25
|
ralbii |
⊢ ( ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ ↔ ∀ 𝑦 ∈ 𝑥 { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑦 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) |
27 |
|
imaeq2 |
⊢ ( 𝑦 = 𝑏 → ( 𝐹 “ 𝑦 ) = ( 𝐹 “ 𝑏 ) ) |
28 |
27
|
raleqdv |
⊢ ( 𝑦 = 𝑏 → ( ∀ 𝑔 ∈ ( 𝐹 “ 𝑦 ) 𝑔 𝑅 𝑧 ↔ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 ) ) |
29 |
28
|
rabbidv |
⊢ ( 𝑦 = 𝑏 → { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑦 ) 𝑔 𝑅 𝑧 } = { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ) |
30 |
29
|
neeq1d |
⊢ ( 𝑦 = 𝑏 → ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑦 ) 𝑔 𝑅 𝑧 } ≠ ∅ ↔ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) |
31 |
30
|
rspccv |
⊢ ( ∀ 𝑦 ∈ 𝑥 { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑦 ) 𝑔 𝑅 𝑧 } ≠ ∅ → ( 𝑏 ∈ 𝑥 → { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) |
32 |
|
imaeq2 |
⊢ ( 𝑦 = 𝑎 → ( 𝐹 “ 𝑦 ) = ( 𝐹 “ 𝑎 ) ) |
33 |
32
|
raleqdv |
⊢ ( 𝑦 = 𝑎 → ( ∀ 𝑔 ∈ ( 𝐹 “ 𝑦 ) 𝑔 𝑅 𝑧 ↔ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 ) ) |
34 |
33
|
rabbidv |
⊢ ( 𝑦 = 𝑎 → { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑦 ) 𝑔 𝑅 𝑧 } = { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ) |
35 |
34
|
neeq1d |
⊢ ( 𝑦 = 𝑎 → ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑦 ) 𝑔 𝑅 𝑧 } ≠ ∅ ↔ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) |
36 |
35
|
rspccv |
⊢ ( ∀ 𝑦 ∈ 𝑥 { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑦 ) 𝑔 𝑅 𝑧 } ≠ ∅ → ( 𝑎 ∈ 𝑥 → { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) |
37 |
31 36
|
anim12d |
⊢ ( ∀ 𝑦 ∈ 𝑥 { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑦 ) 𝑔 𝑅 𝑧 } ≠ ∅ → ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) → ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) ) |
38 |
26 37
|
sylbi |
⊢ ( ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ → ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) → ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) ) |
39 |
|
onelon |
⊢ ( ( 𝑥 ∈ On ∧ 𝑏 ∈ 𝑥 ) → 𝑏 ∈ On ) |
40 |
|
onelon |
⊢ ( ( 𝑥 ∈ On ∧ 𝑎 ∈ 𝑥 ) → 𝑎 ∈ On ) |
41 |
39 40
|
anim12dan |
⊢ ( ( 𝑥 ∈ On ∧ ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) ) → ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) ) |
42 |
41
|
ex |
⊢ ( 𝑥 ∈ On → ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) → ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) ) ) |
43 |
|
eloni |
⊢ ( 𝑏 ∈ On → Ord 𝑏 ) |
44 |
|
eloni |
⊢ ( 𝑎 ∈ On → Ord 𝑎 ) |
45 |
|
ordtri3or |
⊢ ( ( Ord 𝑏 ∧ Ord 𝑎 ) → ( 𝑏 ∈ 𝑎 ∨ 𝑏 = 𝑎 ∨ 𝑎 ∈ 𝑏 ) ) |
46 |
43 44 45
|
syl2an |
⊢ ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) → ( 𝑏 ∈ 𝑎 ∨ 𝑏 = 𝑎 ∨ 𝑎 ∈ 𝑏 ) ) |
47 |
|
eqid |
⊢ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } = { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } |
48 |
1 2 47
|
zorn2lem2 |
⊢ ( ( 𝑎 ∈ On ∧ ( 𝑤 We 𝐴 ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) → ( 𝑏 ∈ 𝑎 → ( 𝐹 ‘ 𝑏 ) 𝑅 ( 𝐹 ‘ 𝑎 ) ) ) |
49 |
48
|
adantll |
⊢ ( ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) ∧ ( 𝑤 We 𝐴 ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) → ( 𝑏 ∈ 𝑎 → ( 𝐹 ‘ 𝑏 ) 𝑅 ( 𝐹 ‘ 𝑎 ) ) ) |
50 |
|
breq12 |
⊢ ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( ( 𝐹 ‘ 𝑏 ) 𝑅 ( 𝐹 ‘ 𝑎 ) ↔ 𝑠 𝑅 𝑟 ) ) |
51 |
50
|
biimpcd |
⊢ ( ( 𝐹 ‘ 𝑏 ) 𝑅 ( 𝐹 ‘ 𝑎 ) → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → 𝑠 𝑅 𝑟 ) ) |
52 |
49 51
|
syl6 |
⊢ ( ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) ∧ ( 𝑤 We 𝐴 ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) → ( 𝑏 ∈ 𝑎 → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → 𝑠 𝑅 𝑟 ) ) ) |
53 |
52
|
com23 |
⊢ ( ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) ∧ ( 𝑤 We 𝐴 ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( 𝑏 ∈ 𝑎 → 𝑠 𝑅 𝑟 ) ) ) |
54 |
53
|
adantrrl |
⊢ ( ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) ∧ ( 𝑤 We 𝐴 ∧ ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) ) → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( 𝑏 ∈ 𝑎 → 𝑠 𝑅 𝑟 ) ) ) |
55 |
54
|
imp |
⊢ ( ( ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) ∧ ( 𝑤 We 𝐴 ∧ ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) ) ∧ ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) → ( 𝑏 ∈ 𝑎 → 𝑠 𝑅 𝑟 ) ) |
56 |
|
fveq2 |
⊢ ( 𝑏 = 𝑎 → ( 𝐹 ‘ 𝑏 ) = ( 𝐹 ‘ 𝑎 ) ) |
57 |
|
eqeq12 |
⊢ ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( ( 𝐹 ‘ 𝑏 ) = ( 𝐹 ‘ 𝑎 ) ↔ 𝑠 = 𝑟 ) ) |
58 |
56 57
|
syl5ib |
⊢ ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( 𝑏 = 𝑎 → 𝑠 = 𝑟 ) ) |
59 |
58
|
adantl |
⊢ ( ( ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) ∧ ( 𝑤 We 𝐴 ∧ ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) ) ∧ ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) → ( 𝑏 = 𝑎 → 𝑠 = 𝑟 ) ) |
60 |
|
eqid |
⊢ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } = { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } |
61 |
1 2 60
|
zorn2lem2 |
⊢ ( ( 𝑏 ∈ On ∧ ( 𝑤 We 𝐴 ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) → ( 𝑎 ∈ 𝑏 → ( 𝐹 ‘ 𝑎 ) 𝑅 ( 𝐹 ‘ 𝑏 ) ) ) |
62 |
61
|
adantlr |
⊢ ( ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) ∧ ( 𝑤 We 𝐴 ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) → ( 𝑎 ∈ 𝑏 → ( 𝐹 ‘ 𝑎 ) 𝑅 ( 𝐹 ‘ 𝑏 ) ) ) |
63 |
|
breq12 |
⊢ ( ( ( 𝐹 ‘ 𝑎 ) = 𝑟 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑠 ) → ( ( 𝐹 ‘ 𝑎 ) 𝑅 ( 𝐹 ‘ 𝑏 ) ↔ 𝑟 𝑅 𝑠 ) ) |
64 |
63
|
ancoms |
⊢ ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( ( 𝐹 ‘ 𝑎 ) 𝑅 ( 𝐹 ‘ 𝑏 ) ↔ 𝑟 𝑅 𝑠 ) ) |
65 |
64
|
biimpcd |
⊢ ( ( 𝐹 ‘ 𝑎 ) 𝑅 ( 𝐹 ‘ 𝑏 ) → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → 𝑟 𝑅 𝑠 ) ) |
66 |
62 65
|
syl6 |
⊢ ( ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) ∧ ( 𝑤 We 𝐴 ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) → ( 𝑎 ∈ 𝑏 → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → 𝑟 𝑅 𝑠 ) ) ) |
67 |
66
|
com23 |
⊢ ( ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) ∧ ( 𝑤 We 𝐴 ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( 𝑎 ∈ 𝑏 → 𝑟 𝑅 𝑠 ) ) ) |
68 |
67
|
adantrrr |
⊢ ( ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) ∧ ( 𝑤 We 𝐴 ∧ ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) ) → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( 𝑎 ∈ 𝑏 → 𝑟 𝑅 𝑠 ) ) ) |
69 |
68
|
imp |
⊢ ( ( ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) ∧ ( 𝑤 We 𝐴 ∧ ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) ) ∧ ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) → ( 𝑎 ∈ 𝑏 → 𝑟 𝑅 𝑠 ) ) |
70 |
55 59 69
|
3orim123d |
⊢ ( ( ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) ∧ ( 𝑤 We 𝐴 ∧ ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) ) ∧ ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) → ( ( 𝑏 ∈ 𝑎 ∨ 𝑏 = 𝑎 ∨ 𝑎 ∈ 𝑏 ) → ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) ) |
71 |
46 70
|
syl5 |
⊢ ( ( ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) ∧ ( 𝑤 We 𝐴 ∧ ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) ) ∧ ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) → ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) → ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) ) |
72 |
71
|
exp31 |
⊢ ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) → ( ( 𝑤 We 𝐴 ∧ ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) → ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) ) ) ) |
73 |
72
|
com4r |
⊢ ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) → ( ( 𝑏 ∈ On ∧ 𝑎 ∈ On ) → ( ( 𝑤 We 𝐴 ∧ ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) ) ) ) |
74 |
42 42 73
|
syl6c |
⊢ ( 𝑥 ∈ On → ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) → ( ( 𝑤 We 𝐴 ∧ ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) ) ) ) |
75 |
74
|
exp4a |
⊢ ( 𝑥 ∈ On → ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) → ( 𝑤 We 𝐴 → ( ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) ) ) ) ) |
76 |
75
|
com3r |
⊢ ( 𝑤 We 𝐴 → ( 𝑥 ∈ On → ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) → ( ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) ) ) ) ) |
77 |
76
|
imp |
⊢ ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) → ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) → ( ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) ) ) ) |
78 |
77
|
a2d |
⊢ ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) → ( ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) → ( { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑏 ) 𝑔 𝑅 𝑧 } ≠ ∅ ∧ { 𝑧 ∈ 𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹 “ 𝑎 ) 𝑔 𝑅 𝑧 } ≠ ∅ ) ) → ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) ) ) ) |
79 |
38 78
|
syl5 |
⊢ ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) → ( ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ → ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) → ( ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) → ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) ) ) ) |
80 |
79
|
imp4b |
⊢ ( ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) ∧ ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ ) → ( ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) ∧ ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) → ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) ) |
81 |
80
|
exlimdvv |
⊢ ( ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) ∧ ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ ) → ( ∃ 𝑏 ∃ 𝑎 ( ( 𝑏 ∈ 𝑥 ∧ 𝑎 ∈ 𝑥 ) ∧ ( ( 𝐹 ‘ 𝑏 ) = 𝑠 ∧ ( 𝐹 ‘ 𝑎 ) = 𝑟 ) ) → ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) ) |
82 |
24 81
|
syl5 |
⊢ ( ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) ∧ ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ ) → ( ( 𝑠 ∈ ( 𝐹 “ 𝑥 ) ∧ 𝑟 ∈ ( 𝐹 “ 𝑥 ) ) → ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) ) |
83 |
82
|
ralrimivv |
⊢ ( ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) ∧ ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ ) → ∀ 𝑠 ∈ ( 𝐹 “ 𝑥 ) ∀ 𝑟 ∈ ( 𝐹 “ 𝑥 ) ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) |
84 |
7 83
|
jca2 |
⊢ ( 𝑅 Po 𝐴 → ( ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) ∧ ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ ) → ( 𝑅 Po ( 𝐹 “ 𝑥 ) ∧ ∀ 𝑠 ∈ ( 𝐹 “ 𝑥 ) ∀ 𝑟 ∈ ( 𝐹 “ 𝑥 ) ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) ) ) |
85 |
|
df-so |
⊢ ( 𝑅 Or ( 𝐹 “ 𝑥 ) ↔ ( 𝑅 Po ( 𝐹 “ 𝑥 ) ∧ ∀ 𝑠 ∈ ( 𝐹 “ 𝑥 ) ∀ 𝑟 ∈ ( 𝐹 “ 𝑥 ) ( 𝑠 𝑅 𝑟 ∨ 𝑠 = 𝑟 ∨ 𝑟 𝑅 𝑠 ) ) ) |
86 |
84 85
|
syl6ibr |
⊢ ( 𝑅 Po 𝐴 → ( ( ( 𝑤 We 𝐴 ∧ 𝑥 ∈ On ) ∧ ∀ 𝑦 ∈ 𝑥 𝐻 ≠ ∅ ) → 𝑅 Or ( 𝐹 “ 𝑥 ) ) ) |