Step |
Hyp |
Ref |
Expression |
1 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝐴 ↔ 𝑦 = 𝐴 ) ) |
2 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝐵 ↔ 𝑦 = 𝐵 ) ) |
3 |
1 2
|
orbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ) ↔ ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) ) ) |
4 |
3
|
reu8 |
⊢ ( ∃! 𝑥 ∈ 𝑉 ( 𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ) ↔ ∃ 𝑥 ∈ 𝑉 ( ( 𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ) ∧ ∀ 𝑦 ∈ 𝑉 ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) → 𝑥 = 𝑦 ) ) ) |
5 |
|
simprlr |
⊢ ( ( 𝑥 = 𝐴 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → 𝐵 ∈ 𝑉 ) |
6 |
|
eqeq1 |
⊢ ( 𝑦 = 𝐵 → ( 𝑦 = 𝐴 ↔ 𝐵 = 𝐴 ) ) |
7 |
|
eqeq1 |
⊢ ( 𝑦 = 𝐵 → ( 𝑦 = 𝐵 ↔ 𝐵 = 𝐵 ) ) |
8 |
6 7
|
orbi12d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) ↔ ( 𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ) ) ) |
9 |
|
eqeq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝐵 ) ) |
10 |
8 9
|
imbi12d |
⊢ ( 𝑦 = 𝐵 → ( ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) → 𝑥 = 𝑦 ) ↔ ( ( 𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ) → 𝑥 = 𝐵 ) ) ) |
11 |
10
|
rspcv |
⊢ ( 𝐵 ∈ 𝑉 → ( ∀ 𝑦 ∈ 𝑉 ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) → 𝑥 = 𝑦 ) → ( ( 𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ) → 𝑥 = 𝐵 ) ) ) |
12 |
5 11
|
syl |
⊢ ( ( 𝑥 = 𝐴 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → ( ∀ 𝑦 ∈ 𝑉 ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) → 𝑥 = 𝑦 ) → ( ( 𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ) → 𝑥 = 𝐵 ) ) ) |
13 |
|
ioran |
⊢ ( ¬ ( 𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ) ↔ ( ¬ 𝐵 = 𝐴 ∧ ¬ 𝐵 = 𝐵 ) ) |
14 |
|
eqid |
⊢ 𝐵 = 𝐵 |
15 |
14
|
pm2.24i |
⊢ ( ¬ 𝐵 = 𝐵 → ( ( 𝑥 = 𝐴 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → 𝐴 = 𝐵 ) ) |
16 |
13 15
|
simplbiim |
⊢ ( ¬ ( 𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ) → ( ( 𝑥 = 𝐴 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → 𝐴 = 𝐵 ) ) |
17 |
|
eqtr2 |
⊢ ( ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐵 ) → 𝐴 = 𝐵 ) |
18 |
17
|
ancoms |
⊢ ( ( 𝑥 = 𝐵 ∧ 𝑥 = 𝐴 ) → 𝐴 = 𝐵 ) |
19 |
18
|
a1d |
⊢ ( ( 𝑥 = 𝐵 ∧ 𝑥 = 𝐴 ) → ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) → 𝐴 = 𝐵 ) ) |
20 |
19
|
expimpd |
⊢ ( 𝑥 = 𝐵 → ( ( 𝑥 = 𝐴 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → 𝐴 = 𝐵 ) ) |
21 |
16 20
|
ja |
⊢ ( ( ( 𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ) → 𝑥 = 𝐵 ) → ( ( 𝑥 = 𝐴 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → 𝐴 = 𝐵 ) ) |
22 |
21
|
com12 |
⊢ ( ( 𝑥 = 𝐴 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → ( ( ( 𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ) → 𝑥 = 𝐵 ) → 𝐴 = 𝐵 ) ) |
23 |
12 22
|
syld |
⊢ ( ( 𝑥 = 𝐴 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → ( ∀ 𝑦 ∈ 𝑉 ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) → 𝑥 = 𝑦 ) → 𝐴 = 𝐵 ) ) |
24 |
23
|
ex |
⊢ ( 𝑥 = 𝐴 → ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) → ( ∀ 𝑦 ∈ 𝑉 ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) → 𝑥 = 𝑦 ) → 𝐴 = 𝐵 ) ) ) |
25 |
|
simprll |
⊢ ( ( 𝑥 = 𝐵 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → 𝐴 ∈ 𝑉 ) |
26 |
|
eqeq1 |
⊢ ( 𝑦 = 𝐴 → ( 𝑦 = 𝐴 ↔ 𝐴 = 𝐴 ) ) |
27 |
|
eqeq1 |
⊢ ( 𝑦 = 𝐴 → ( 𝑦 = 𝐵 ↔ 𝐴 = 𝐵 ) ) |
28 |
26 27
|
orbi12d |
⊢ ( 𝑦 = 𝐴 → ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) ↔ ( 𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ) ) ) |
29 |
|
eqeq2 |
⊢ ( 𝑦 = 𝐴 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝐴 ) ) |
30 |
28 29
|
imbi12d |
⊢ ( 𝑦 = 𝐴 → ( ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) → 𝑥 = 𝑦 ) ↔ ( ( 𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ) → 𝑥 = 𝐴 ) ) ) |
31 |
30
|
rspcv |
⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑦 ∈ 𝑉 ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) → 𝑥 = 𝑦 ) → ( ( 𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ) → 𝑥 = 𝐴 ) ) ) |
32 |
25 31
|
syl |
⊢ ( ( 𝑥 = 𝐵 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → ( ∀ 𝑦 ∈ 𝑉 ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) → 𝑥 = 𝑦 ) → ( ( 𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ) → 𝑥 = 𝐴 ) ) ) |
33 |
|
ioran |
⊢ ( ¬ ( 𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ) ↔ ( ¬ 𝐴 = 𝐴 ∧ ¬ 𝐴 = 𝐵 ) ) |
34 |
|
eqid |
⊢ 𝐴 = 𝐴 |
35 |
34
|
pm2.24i |
⊢ ( ¬ 𝐴 = 𝐴 → ( ( 𝑥 = 𝐵 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → 𝐴 = 𝐵 ) ) |
36 |
35
|
adantr |
⊢ ( ( ¬ 𝐴 = 𝐴 ∧ ¬ 𝐴 = 𝐵 ) → ( ( 𝑥 = 𝐵 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → 𝐴 = 𝐵 ) ) |
37 |
33 36
|
sylbi |
⊢ ( ¬ ( 𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ) → ( ( 𝑥 = 𝐵 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → 𝐴 = 𝐵 ) ) |
38 |
17
|
a1d |
⊢ ( ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐵 ) → ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) → 𝐴 = 𝐵 ) ) |
39 |
38
|
expimpd |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 = 𝐵 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → 𝐴 = 𝐵 ) ) |
40 |
37 39
|
ja |
⊢ ( ( ( 𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ) → 𝑥 = 𝐴 ) → ( ( 𝑥 = 𝐵 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → 𝐴 = 𝐵 ) ) |
41 |
40
|
com12 |
⊢ ( ( 𝑥 = 𝐵 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → ( ( ( 𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ) → 𝑥 = 𝐴 ) → 𝐴 = 𝐵 ) ) |
42 |
32 41
|
syld |
⊢ ( ( 𝑥 = 𝐵 ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) ) → ( ∀ 𝑦 ∈ 𝑉 ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) → 𝑥 = 𝑦 ) → 𝐴 = 𝐵 ) ) |
43 |
42
|
ex |
⊢ ( 𝑥 = 𝐵 → ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) → ( ∀ 𝑦 ∈ 𝑉 ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) → 𝑥 = 𝑦 ) → 𝐴 = 𝐵 ) ) ) |
44 |
24 43
|
jaoi |
⊢ ( ( 𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ) → ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) → ( ∀ 𝑦 ∈ 𝑉 ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) → 𝑥 = 𝑦 ) → 𝐴 = 𝐵 ) ) ) |
45 |
44
|
com12 |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) → ( ( 𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ) → ( ∀ 𝑦 ∈ 𝑉 ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) → 𝑥 = 𝑦 ) → 𝐴 = 𝐵 ) ) ) |
46 |
45
|
impd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝑉 ) → ( ( ( 𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ) ∧ ∀ 𝑦 ∈ 𝑉 ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) → 𝑥 = 𝑦 ) ) → 𝐴 = 𝐵 ) ) |
47 |
46
|
rexlimdva |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ∃ 𝑥 ∈ 𝑉 ( ( 𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ) ∧ ∀ 𝑦 ∈ 𝑉 ( ( 𝑦 = 𝐴 ∨ 𝑦 = 𝐵 ) → 𝑥 = 𝑦 ) ) → 𝐴 = 𝐵 ) ) |
48 |
4 47
|
syl5bi |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ∃! 𝑥 ∈ 𝑉 ( 𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ) → 𝐴 = 𝐵 ) ) |
49 |
|
reueq |
⊢ ( 𝐵 ∈ 𝑉 ↔ ∃! 𝑥 ∈ 𝑉 𝑥 = 𝐵 ) |
50 |
49
|
biimpi |
⊢ ( 𝐵 ∈ 𝑉 → ∃! 𝑥 ∈ 𝑉 𝑥 = 𝐵 ) |
51 |
50
|
adantl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ∃! 𝑥 ∈ 𝑉 𝑥 = 𝐵 ) |
52 |
51
|
adantr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝐴 = 𝐵 ) → ∃! 𝑥 ∈ 𝑉 𝑥 = 𝐵 ) |
53 |
|
eqeq2 |
⊢ ( 𝐴 = 𝐵 → ( 𝑥 = 𝐴 ↔ 𝑥 = 𝐵 ) ) |
54 |
53
|
adantl |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝐴 = 𝐵 ) → ( 𝑥 = 𝐴 ↔ 𝑥 = 𝐵 ) ) |
55 |
54
|
orbi1d |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝐴 = 𝐵 ) → ( ( 𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ) ↔ ( 𝑥 = 𝐵 ∨ 𝑥 = 𝐵 ) ) ) |
56 |
|
oridm |
⊢ ( ( 𝑥 = 𝐵 ∨ 𝑥 = 𝐵 ) ↔ 𝑥 = 𝐵 ) |
57 |
55 56
|
bitrdi |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝐴 = 𝐵 ) → ( ( 𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ) ↔ 𝑥 = 𝐵 ) ) |
58 |
57
|
reubidv |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝐴 = 𝐵 ) → ( ∃! 𝑥 ∈ 𝑉 ( 𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ) ↔ ∃! 𝑥 ∈ 𝑉 𝑥 = 𝐵 ) ) |
59 |
52 58
|
mpbird |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝐴 = 𝐵 ) → ∃! 𝑥 ∈ 𝑉 ( 𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ) ) |
60 |
59
|
ex |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( 𝐴 = 𝐵 → ∃! 𝑥 ∈ 𝑉 ( 𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ) ) ) |
61 |
48 60
|
impbid |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ∃! 𝑥 ∈ 𝑉 ( 𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ) ↔ 𝐴 = 𝐵 ) ) |