Step |
Hyp |
Ref |
Expression |
1 |
|
ordtypelem.1 |
⊢ 𝐹 = recs ( 𝐺 ) |
2 |
|
ordtypelem.2 |
⊢ 𝐶 = { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } |
3 |
|
ordtypelem.3 |
⊢ 𝐺 = ( ℎ ∈ V ↦ ( ℩ 𝑣 ∈ 𝐶 ∀ 𝑢 ∈ 𝐶 ¬ 𝑢 𝑅 𝑣 ) ) |
4 |
|
ordtypelem.5 |
⊢ 𝑇 = { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 } |
5 |
|
ordtypelem.6 |
⊢ 𝑂 = OrdIso ( 𝑅 , 𝐴 ) |
6 |
|
ordtypelem.7 |
⊢ ( 𝜑 → 𝑅 We 𝐴 ) |
7 |
|
ordtypelem.8 |
⊢ ( 𝜑 → 𝑅 Se 𝐴 ) |
8 |
4
|
ssrab3 |
⊢ 𝑇 ⊆ On |
9 |
8
|
a1i |
⊢ ( 𝜑 → 𝑇 ⊆ On ) |
10 |
9
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑇 ) → 𝑎 ∈ On ) |
11 |
|
onss |
⊢ ( 𝑎 ∈ On → 𝑎 ⊆ On ) |
12 |
10 11
|
syl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑇 ) → 𝑎 ⊆ On ) |
13 |
|
eloni |
⊢ ( 𝑎 ∈ On → Ord 𝑎 ) |
14 |
10 13
|
syl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑇 ) → Ord 𝑎 ) |
15 |
|
imaeq2 |
⊢ ( 𝑥 = 𝑎 → ( 𝐹 “ 𝑥 ) = ( 𝐹 “ 𝑎 ) ) |
16 |
15
|
raleqdv |
⊢ ( 𝑥 = 𝑎 → ( ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 ↔ ∀ 𝑧 ∈ ( 𝐹 “ 𝑎 ) 𝑧 𝑅 𝑡 ) ) |
17 |
16
|
rexbidv |
⊢ ( 𝑥 = 𝑎 → ( ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 ↔ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑎 ) 𝑧 𝑅 𝑡 ) ) |
18 |
17 4
|
elrab2 |
⊢ ( 𝑎 ∈ 𝑇 ↔ ( 𝑎 ∈ On ∧ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑎 ) 𝑧 𝑅 𝑡 ) ) |
19 |
18
|
simprbi |
⊢ ( 𝑎 ∈ 𝑇 → ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑎 ) 𝑧 𝑅 𝑡 ) |
20 |
19
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑇 ) → ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑎 ) 𝑧 𝑅 𝑡 ) |
21 |
|
ordelss |
⊢ ( ( Ord 𝑎 ∧ 𝑥 ∈ 𝑎 ) → 𝑥 ⊆ 𝑎 ) |
22 |
|
imass2 |
⊢ ( 𝑥 ⊆ 𝑎 → ( 𝐹 “ 𝑥 ) ⊆ ( 𝐹 “ 𝑎 ) ) |
23 |
|
ssralv |
⊢ ( ( 𝐹 “ 𝑥 ) ⊆ ( 𝐹 “ 𝑎 ) → ( ∀ 𝑧 ∈ ( 𝐹 “ 𝑎 ) 𝑧 𝑅 𝑡 → ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 ) ) |
24 |
23
|
reximdv |
⊢ ( ( 𝐹 “ 𝑥 ) ⊆ ( 𝐹 “ 𝑎 ) → ( ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑎 ) 𝑧 𝑅 𝑡 → ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 ) ) |
25 |
21 22 24
|
3syl |
⊢ ( ( Ord 𝑎 ∧ 𝑥 ∈ 𝑎 ) → ( ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑎 ) 𝑧 𝑅 𝑡 → ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 ) ) |
26 |
25
|
ralrimdva |
⊢ ( Ord 𝑎 → ( ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑎 ) 𝑧 𝑅 𝑡 → ∀ 𝑥 ∈ 𝑎 ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 ) ) |
27 |
14 20 26
|
sylc |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑇 ) → ∀ 𝑥 ∈ 𝑎 ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 ) |
28 |
|
ssrab |
⊢ ( 𝑎 ⊆ { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 } ↔ ( 𝑎 ⊆ On ∧ ∀ 𝑥 ∈ 𝑎 ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 ) ) |
29 |
12 27 28
|
sylanbrc |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑇 ) → 𝑎 ⊆ { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 } ) |
30 |
29 4
|
sseqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑇 ) → 𝑎 ⊆ 𝑇 ) |
31 |
30
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝑇 𝑎 ⊆ 𝑇 ) |
32 |
|
dftr3 |
⊢ ( Tr 𝑇 ↔ ∀ 𝑎 ∈ 𝑇 𝑎 ⊆ 𝑇 ) |
33 |
31 32
|
sylibr |
⊢ ( 𝜑 → Tr 𝑇 ) |
34 |
|
ordon |
⊢ Ord On |
35 |
|
trssord |
⊢ ( ( Tr 𝑇 ∧ 𝑇 ⊆ On ∧ Ord On ) → Ord 𝑇 ) |
36 |
8 34 35
|
mp3an23 |
⊢ ( Tr 𝑇 → Ord 𝑇 ) |
37 |
33 36
|
syl |
⊢ ( 𝜑 → Ord 𝑇 ) |