Step |
Hyp |
Ref |
Expression |
1 |
|
weiun.1 |
⊢ 𝐹 = ( 𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ ( ℩ 𝑢 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑢 ) ) |
2 |
|
weiun.2 |
⊢ 𝑇 = { 〈 𝑦 , 𝑧 〉 ∣ ( ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ∧ ( ( 𝐹 ‘ 𝑦 ) 𝑅 ( 𝐹 ‘ 𝑧 ) ∨ ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ∧ 𝑦 ⦋ ( 𝐹 ‘ 𝑦 ) / 𝑥 ⦌ 𝑆 𝑧 ) ) ) } |
3 |
|
sopo |
⊢ ( 𝑆 Or 𝐵 → 𝑆 Po 𝐵 ) |
4 |
3
|
ralimi |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 → ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) |
5 |
1 2
|
weiunpo |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) → 𝑇 Po ∪ 𝑥 ∈ 𝐴 𝐵 ) |
6 |
4 5
|
syl3an3 |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) → 𝑇 Po ∪ 𝑥 ∈ 𝐴 𝐵 ) |
7 |
|
simplrl |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) → 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
8 |
|
simplrr |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) → 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
9 |
|
animorrl |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) → ( ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) |
10 |
1 2
|
weiunlem1 |
⊢ ( 𝑞 𝑇 𝑟 ↔ ( ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ∧ ( ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) ) |
11 |
7 8 9 10
|
syl21anbrc |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) → 𝑞 𝑇 𝑟 ) |
12 |
11
|
3mix1d |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) → ( 𝑞 𝑇 𝑟 ∨ 𝑞 = 𝑟 ∨ 𝑟 𝑇 𝑞 ) ) |
13 |
|
csbeq1 |
⊢ ( 𝑠 = ( 𝐹 ‘ 𝑞 ) → ⦋ 𝑠 / 𝑥 ⦌ 𝑆 = ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 ) |
14 |
|
csbeq1 |
⊢ ( 𝑠 = ( 𝐹 ‘ 𝑞 ) → ⦋ 𝑠 / 𝑥 ⦌ 𝐵 = ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 ) |
15 |
13 14
|
soeq12d |
⊢ ( 𝑠 = ( 𝐹 ‘ 𝑞 ) → ( ⦋ 𝑠 / 𝑥 ⦌ 𝑆 Or ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ↔ ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 Or ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 ) ) |
16 |
|
simpll3 |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) |
17 |
|
nfv |
⊢ Ⅎ 𝑠 𝑆 Or 𝐵 |
18 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑠 / 𝑥 ⦌ 𝑆 |
19 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑠 / 𝑥 ⦌ 𝐵 |
20 |
18 19
|
nfso |
⊢ Ⅎ 𝑥 ⦋ 𝑠 / 𝑥 ⦌ 𝑆 Or ⦋ 𝑠 / 𝑥 ⦌ 𝐵 |
21 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑠 → 𝑆 = ⦋ 𝑠 / 𝑥 ⦌ 𝑆 ) |
22 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑠 → 𝐵 = ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) |
23 |
21 22
|
soeq12d |
⊢ ( 𝑥 = 𝑠 → ( 𝑆 Or 𝐵 ↔ ⦋ 𝑠 / 𝑥 ⦌ 𝑆 Or ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) |
24 |
17 20 23
|
cbvralw |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ↔ ∀ 𝑠 ∈ 𝐴 ⦋ 𝑠 / 𝑥 ⦌ 𝑆 Or ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) |
25 |
16 24
|
sylib |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → ∀ 𝑠 ∈ 𝐴 ⦋ 𝑠 / 𝑥 ⦌ 𝑆 Or ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) |
26 |
|
simpl1 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑅 We 𝐴 ) |
27 |
|
simpl2 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑅 Se 𝐴 ) |
28 |
1 2 26 27
|
weiunlem2 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝐹 : ∪ 𝑥 ∈ 𝐴 𝐵 ⟶ 𝐴 ∧ ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ∧ ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) ) |
29 |
28
|
simp1d |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝐹 : ∪ 𝑥 ∈ 𝐴 𝐵 ⟶ 𝐴 ) |
30 |
|
simprl |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
31 |
29 30
|
ffvelcdmd |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝐹 ‘ 𝑞 ) ∈ 𝐴 ) |
32 |
31
|
adantr |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → ( 𝐹 ‘ 𝑞 ) ∈ 𝐴 ) |
33 |
15 25 32
|
rspcdva |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 Or ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 ) |
34 |
|
id |
⊢ ( 𝑡 = 𝑞 → 𝑡 = 𝑞 ) |
35 |
|
fveq2 |
⊢ ( 𝑡 = 𝑞 → ( 𝐹 ‘ 𝑡 ) = ( 𝐹 ‘ 𝑞 ) ) |
36 |
35
|
csbeq1d |
⊢ ( 𝑡 = 𝑞 → ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 = ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 ) |
37 |
34 36
|
eleq12d |
⊢ ( 𝑡 = 𝑞 → ( 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ↔ 𝑞 ∈ ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 ) ) |
38 |
28
|
simp2d |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ) |
39 |
38
|
adantr |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ) |
40 |
|
simplrl |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
41 |
37 39 40
|
rspcdva |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → 𝑞 ∈ ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 ) |
42 |
|
id |
⊢ ( 𝑡 = 𝑟 → 𝑡 = 𝑟 ) |
43 |
|
fveq2 |
⊢ ( 𝑡 = 𝑟 → ( 𝐹 ‘ 𝑡 ) = ( 𝐹 ‘ 𝑟 ) ) |
44 |
43
|
csbeq1d |
⊢ ( 𝑡 = 𝑟 → ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 = ⦋ ( 𝐹 ‘ 𝑟 ) / 𝑥 ⦌ 𝐵 ) |
45 |
42 44
|
eleq12d |
⊢ ( 𝑡 = 𝑟 → ( 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ↔ 𝑟 ∈ ⦋ ( 𝐹 ‘ 𝑟 ) / 𝑥 ⦌ 𝐵 ) ) |
46 |
|
simplrr |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
47 |
45 39 46
|
rspcdva |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → 𝑟 ∈ ⦋ ( 𝐹 ‘ 𝑟 ) / 𝑥 ⦌ 𝐵 ) |
48 |
|
simpr |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) |
49 |
48
|
csbeq1d |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 = ⦋ ( 𝐹 ‘ 𝑟 ) / 𝑥 ⦌ 𝐵 ) |
50 |
47 49
|
eleqtrrd |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → 𝑟 ∈ ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 ) |
51 |
|
solin |
⊢ ( ( ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 Or ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 ∧ ( 𝑞 ∈ ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 ∧ 𝑟 ∈ ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 ) ) → ( 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ∨ 𝑞 = 𝑟 ∨ 𝑟 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑞 ) ) |
52 |
33 41 50 51
|
syl12anc |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → ( 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ∨ 𝑞 = 𝑟 ∨ 𝑟 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑞 ) ) |
53 |
|
simpllr |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) → ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) |
54 |
48
|
anim1i |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) → ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) |
55 |
54
|
olcd |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) → ( ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) |
56 |
53 55 10
|
sylanbrc |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) → 𝑞 𝑇 𝑟 ) |
57 |
56
|
ex |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → ( 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 → 𝑞 𝑇 𝑟 ) ) |
58 |
|
idd |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → ( 𝑞 = 𝑟 → 𝑞 = 𝑟 ) ) |
59 |
46
|
adantr |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) ∧ 𝑟 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑞 ) → 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
60 |
40
|
adantr |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) ∧ 𝑟 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑞 ) → 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
61 |
|
simplr |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) ∧ 𝑟 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑞 ) → ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) |
62 |
61
|
eqcomd |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) ∧ 𝑟 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑞 ) → ( 𝐹 ‘ 𝑟 ) = ( 𝐹 ‘ 𝑞 ) ) |
63 |
61
|
csbeq1d |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) ∧ 𝑟 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑞 ) → ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 = ⦋ ( 𝐹 ‘ 𝑟 ) / 𝑥 ⦌ 𝑆 ) |
64 |
|
simpr |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) ∧ 𝑟 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑞 ) → 𝑟 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑞 ) |
65 |
63 64
|
breqdi |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) ∧ 𝑟 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑞 ) → 𝑟 ⦋ ( 𝐹 ‘ 𝑟 ) / 𝑥 ⦌ 𝑆 𝑞 ) |
66 |
62 65
|
jca |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) ∧ 𝑟 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑞 ) → ( ( 𝐹 ‘ 𝑟 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑟 ⦋ ( 𝐹 ‘ 𝑟 ) / 𝑥 ⦌ 𝑆 𝑞 ) ) |
67 |
66
|
olcd |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) ∧ 𝑟 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑞 ) → ( ( 𝐹 ‘ 𝑟 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ∨ ( ( 𝐹 ‘ 𝑟 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑟 ⦋ ( 𝐹 ‘ 𝑟 ) / 𝑥 ⦌ 𝑆 𝑞 ) ) ) |
68 |
1 2
|
weiunlem1 |
⊢ ( 𝑟 𝑇 𝑞 ↔ ( ( 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ∧ ( ( 𝐹 ‘ 𝑟 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ∨ ( ( 𝐹 ‘ 𝑟 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑟 ⦋ ( 𝐹 ‘ 𝑟 ) / 𝑥 ⦌ 𝑆 𝑞 ) ) ) ) |
69 |
59 60 67 68
|
syl21anbrc |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) ∧ 𝑟 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑞 ) → 𝑟 𝑇 𝑞 ) |
70 |
69
|
ex |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → ( 𝑟 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑞 → 𝑟 𝑇 𝑞 ) ) |
71 |
57 58 70
|
3orim123d |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → ( ( 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ∨ 𝑞 = 𝑟 ∨ 𝑟 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑞 ) → ( 𝑞 𝑇 𝑟 ∨ 𝑞 = 𝑟 ∨ 𝑟 𝑇 𝑞 ) ) ) |
72 |
52 71
|
mpd |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) → ( 𝑞 𝑇 𝑟 ∨ 𝑞 = 𝑟 ∨ 𝑟 𝑇 𝑞 ) ) |
73 |
|
simplrr |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑟 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ) → 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
74 |
|
simplrl |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑟 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ) → 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
75 |
|
animorrl |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑟 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ) → ( ( 𝐹 ‘ 𝑟 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ∨ ( ( 𝐹 ‘ 𝑟 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑟 ⦋ ( 𝐹 ‘ 𝑟 ) / 𝑥 ⦌ 𝑆 𝑞 ) ) ) |
76 |
73 74 75 68
|
syl21anbrc |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑟 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ) → 𝑟 𝑇 𝑞 ) |
77 |
76
|
3mix3d |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( 𝐹 ‘ 𝑟 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ) → ( 𝑞 𝑇 𝑟 ∨ 𝑞 = 𝑟 ∨ 𝑟 𝑇 𝑞 ) ) |
78 |
|
weso |
⊢ ( 𝑅 We 𝐴 → 𝑅 Or 𝐴 ) |
79 |
26 78
|
syl |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑅 Or 𝐴 ) |
80 |
|
simprr |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
81 |
29 80
|
ffvelcdmd |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝐹 ‘ 𝑟 ) ∈ 𝐴 ) |
82 |
|
solin |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( ( 𝐹 ‘ 𝑞 ) ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑟 ) ∈ 𝐴 ) ) → ( ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∨ ( 𝐹 ‘ 𝑟 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ) ) |
83 |
79 31 81 82
|
syl12anc |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∨ ( 𝐹 ‘ 𝑟 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ) ) |
84 |
12 72 77 83
|
mpjao3dan |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) ∧ ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝑞 𝑇 𝑟 ∨ 𝑞 = 𝑟 ∨ 𝑟 𝑇 𝑞 ) ) |
85 |
6 84
|
issod |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Or 𝐵 ) → 𝑇 Or ∪ 𝑥 ∈ 𝐴 𝐵 ) |