Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) → 𝑅 FrSe 𝐴 ) |
2 |
|
bnj1127 |
⊢ ( 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → 𝑌 ∈ 𝐴 ) |
3 |
2
|
3ad2ant3 |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) → 𝑌 ∈ 𝐴 ) |
4 |
|
bnj893 |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) → trCl ( 𝑋 , 𝐴 , 𝑅 ) ∈ V ) |
5 |
4
|
3adant3 |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) → trCl ( 𝑋 , 𝐴 , 𝑅 ) ∈ V ) |
6 |
|
bnj1029 |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) → TrFo ( trCl ( 𝑋 , 𝐴 , 𝑅 ) , 𝐴 , 𝑅 ) ) |
7 |
6
|
3adant3 |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) → TrFo ( trCl ( 𝑋 , 𝐴 , 𝑅 ) , 𝐴 , 𝑅 ) ) |
8 |
|
simp3 |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) → 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) |
9 |
|
elisset |
⊢ ( 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → ∃ 𝑦 𝑦 = 𝑌 ) |
10 |
9
|
3ad2ant3 |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) → ∃ 𝑦 𝑦 = 𝑌 ) |
11 |
|
df-bnj19 |
⊢ ( TrFo ( trCl ( 𝑋 , 𝐴 , 𝑅 ) , 𝐴 , 𝑅 ) ↔ ∀ 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) |
12 |
|
rsp |
⊢ ( ∀ 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) → ( 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → pred ( 𝑦 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ) |
13 |
11 12
|
sylbi |
⊢ ( TrFo ( trCl ( 𝑋 , 𝐴 , 𝑅 ) , 𝐴 , 𝑅 ) → ( 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → pred ( 𝑦 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ) |
14 |
7 13
|
syl |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) → ( 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → pred ( 𝑦 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ) |
15 |
|
eleq1 |
⊢ ( 𝑦 = 𝑌 → ( 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ↔ 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ) |
16 |
|
bnj602 |
⊢ ( 𝑦 = 𝑌 → pred ( 𝑦 , 𝐴 , 𝑅 ) = pred ( 𝑌 , 𝐴 , 𝑅 ) ) |
17 |
16
|
sseq1d |
⊢ ( 𝑦 = 𝑌 → ( pred ( 𝑦 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ↔ pred ( 𝑌 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ) |
18 |
15 17
|
imbi12d |
⊢ ( 𝑦 = 𝑌 → ( ( 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → pred ( 𝑦 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ↔ ( 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → pred ( 𝑌 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ) ) |
19 |
14 18
|
syl5ib |
⊢ ( 𝑦 = 𝑌 → ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) → ( 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → pred ( 𝑌 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ) ) |
20 |
19
|
exlimiv |
⊢ ( ∃ 𝑦 𝑦 = 𝑌 → ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) → ( 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → pred ( 𝑌 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ) ) |
21 |
10 20
|
mpcom |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) → ( 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → pred ( 𝑌 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ) |
22 |
8 21
|
mpd |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) → pred ( 𝑌 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) |
23 |
|
biid |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑌 ∈ 𝐴 ) ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) |
24 |
|
biid |
⊢ ( ( trCl ( 𝑋 , 𝐴 , 𝑅 ) ∈ V ∧ TrFo ( trCl ( 𝑋 , 𝐴 , 𝑅 ) , 𝐴 , 𝑅 ) ∧ pred ( 𝑌 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ↔ ( trCl ( 𝑋 , 𝐴 , 𝑅 ) ∈ V ∧ TrFo ( trCl ( 𝑋 , 𝐴 , 𝑅 ) , 𝐴 , 𝑅 ) ∧ pred ( 𝑌 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ) |
25 |
23 24
|
bnj1124 |
⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑌 ∈ 𝐴 ) ∧ ( trCl ( 𝑋 , 𝐴 , 𝑅 ) ∈ V ∧ TrFo ( trCl ( 𝑋 , 𝐴 , 𝑅 ) , 𝐴 , 𝑅 ) ∧ pred ( 𝑌 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) ) → trCl ( 𝑌 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) |
26 |
1 3 5 7 22 25
|
syl23anc |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) → trCl ( 𝑌 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) |