Step |
Hyp |
Ref |
Expression |
1 |
|
eluni |
⊢ ( 𝑢 ∈ ∪ 𝐴 ↔ ∃ 𝑦 ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) |
2 |
1
|
anbi2i |
⊢ ( ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴 ) ↔ ( 𝑧 ∈ 𝑢 ∧ ∃ 𝑦 ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
3 |
2
|
exbii |
⊢ ( ∃ 𝑢 ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴 ) ↔ ∃ 𝑢 ( 𝑧 ∈ 𝑢 ∧ ∃ 𝑦 ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
4 |
|
19.42v |
⊢ ( ∃ 𝑦 ( 𝑧 ∈ 𝑢 ∧ ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ↔ ( 𝑧 ∈ 𝑢 ∧ ∃ 𝑦 ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
5 |
4
|
bicomi |
⊢ ( ( 𝑧 ∈ 𝑢 ∧ ∃ 𝑦 ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ↔ ∃ 𝑦 ( 𝑧 ∈ 𝑢 ∧ ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
6 |
5
|
exbii |
⊢ ( ∃ 𝑢 ( 𝑧 ∈ 𝑢 ∧ ∃ 𝑦 ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ↔ ∃ 𝑢 ∃ 𝑦 ( 𝑧 ∈ 𝑢 ∧ ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
7 |
|
excom |
⊢ ( ∃ 𝑢 ∃ 𝑦 ( 𝑧 ∈ 𝑢 ∧ ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ↔ ∃ 𝑦 ∃ 𝑢 ( 𝑧 ∈ 𝑢 ∧ ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
8 |
|
anass |
⊢ ( ( ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦 ) ∧ 𝑦 ∈ 𝐴 ) ↔ ( 𝑧 ∈ 𝑢 ∧ ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
9 |
|
ancom |
⊢ ( ( ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦 ) ∧ 𝑦 ∈ 𝐴 ) ↔ ( 𝑦 ∈ 𝐴 ∧ ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦 ) ) ) |
10 |
8 9
|
bitr3i |
⊢ ( ( 𝑧 ∈ 𝑢 ∧ ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ↔ ( 𝑦 ∈ 𝐴 ∧ ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦 ) ) ) |
11 |
10
|
2exbii |
⊢ ( ∃ 𝑦 ∃ 𝑢 ( 𝑧 ∈ 𝑢 ∧ ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ↔ ∃ 𝑦 ∃ 𝑢 ( 𝑦 ∈ 𝐴 ∧ ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦 ) ) ) |
12 |
|
exdistr |
⊢ ( ∃ 𝑦 ∃ 𝑢 ( 𝑦 ∈ 𝐴 ∧ ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦 ) ) ↔ ∃ 𝑦 ( 𝑦 ∈ 𝐴 ∧ ∃ 𝑢 ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦 ) ) ) |
13 |
7 11 12
|
3bitri |
⊢ ( ∃ 𝑢 ∃ 𝑦 ( 𝑧 ∈ 𝑢 ∧ ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ↔ ∃ 𝑦 ( 𝑦 ∈ 𝐴 ∧ ∃ 𝑢 ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦 ) ) ) |
14 |
|
eluni |
⊢ ( 𝑧 ∈ ∪ 𝑦 ↔ ∃ 𝑢 ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦 ) ) |
15 |
14
|
bicomi |
⊢ ( ∃ 𝑢 ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦 ) ↔ 𝑧 ∈ ∪ 𝑦 ) |
16 |
15
|
anbi2i |
⊢ ( ( 𝑦 ∈ 𝐴 ∧ ∃ 𝑢 ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦 ) ) ↔ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦 ) ) |
17 |
16
|
exbii |
⊢ ( ∃ 𝑦 ( 𝑦 ∈ 𝐴 ∧ ∃ 𝑢 ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦 ) ) ↔ ∃ 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦 ) ) |
18 |
6 13 17
|
3bitri |
⊢ ( ∃ 𝑢 ( 𝑧 ∈ 𝑢 ∧ ∃ 𝑦 ( 𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ↔ ∃ 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦 ) ) |
19 |
|
vuniex |
⊢ ∪ 𝑦 ∈ V |
20 |
|
eleq2 |
⊢ ( 𝑣 = ∪ 𝑦 → ( 𝑧 ∈ 𝑣 ↔ 𝑧 ∈ ∪ 𝑦 ) ) |
21 |
19 20
|
ceqsexv |
⊢ ( ∃ 𝑣 ( 𝑣 = ∪ 𝑦 ∧ 𝑧 ∈ 𝑣 ) ↔ 𝑧 ∈ ∪ 𝑦 ) |
22 |
|
exancom |
⊢ ( ∃ 𝑣 ( 𝑣 = ∪ 𝑦 ∧ 𝑧 ∈ 𝑣 ) ↔ ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦 ) ) |
23 |
21 22
|
bitr3i |
⊢ ( 𝑧 ∈ ∪ 𝑦 ↔ ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦 ) ) |
24 |
23
|
anbi2i |
⊢ ( ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦 ) ↔ ( 𝑦 ∈ 𝐴 ∧ ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦 ) ) ) |
25 |
|
19.42v |
⊢ ( ∃ 𝑣 ( 𝑦 ∈ 𝐴 ∧ ( 𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦 ) ) ↔ ( 𝑦 ∈ 𝐴 ∧ ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦 ) ) ) |
26 |
|
ancom |
⊢ ( ( 𝑦 ∈ 𝐴 ∧ ( 𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦 ) ) ↔ ( ( 𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦 ) ∧ 𝑦 ∈ 𝐴 ) ) |
27 |
|
anass |
⊢ ( ( ( 𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦 ) ∧ 𝑦 ∈ 𝐴 ) ↔ ( 𝑧 ∈ 𝑣 ∧ ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
28 |
26 27
|
bitri |
⊢ ( ( 𝑦 ∈ 𝐴 ∧ ( 𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦 ) ) ↔ ( 𝑧 ∈ 𝑣 ∧ ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
29 |
28
|
exbii |
⊢ ( ∃ 𝑣 ( 𝑦 ∈ 𝐴 ∧ ( 𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦 ) ) ↔ ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
30 |
24 25 29
|
3bitr2i |
⊢ ( ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦 ) ↔ ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
31 |
30
|
exbii |
⊢ ( ∃ 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦 ) ↔ ∃ 𝑦 ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
32 |
|
excom |
⊢ ( ∃ 𝑦 ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ↔ ∃ 𝑣 ∃ 𝑦 ( 𝑧 ∈ 𝑣 ∧ ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
33 |
|
exdistr |
⊢ ( ∃ 𝑣 ∃ 𝑦 ( 𝑧 ∈ 𝑣 ∧ ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ↔ ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ ∃ 𝑦 ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
34 |
|
vex |
⊢ 𝑣 ∈ V |
35 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑣 → ( 𝑥 = ∪ 𝑦 ↔ 𝑣 = ∪ 𝑦 ) ) |
36 |
35
|
anbi1d |
⊢ ( 𝑥 = 𝑣 → ( ( 𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ↔ ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
37 |
36
|
exbidv |
⊢ ( 𝑥 = 𝑣 → ( ∃ 𝑦 ( 𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ↔ ∃ 𝑦 ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ) |
38 |
34 37
|
elab |
⊢ ( 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) } ↔ ∃ 𝑦 ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) |
39 |
38
|
bicomi |
⊢ ( ∃ 𝑦 ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ↔ 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) } ) |
40 |
39
|
anbi2i |
⊢ ( ( 𝑧 ∈ 𝑣 ∧ ∃ 𝑦 ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ↔ ( 𝑧 ∈ 𝑣 ∧ 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) } ) ) |
41 |
40
|
exbii |
⊢ ( ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ ∃ 𝑦 ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ↔ ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) } ) ) |
42 |
33 41
|
bitri |
⊢ ( ∃ 𝑣 ∃ 𝑦 ( 𝑧 ∈ 𝑣 ∧ ( 𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) ↔ ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) } ) ) |
43 |
31 32 42
|
3bitri |
⊢ ( ∃ 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦 ) ↔ ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) } ) ) |
44 |
3 18 43
|
3bitri |
⊢ ( ∃ 𝑢 ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴 ) ↔ ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) } ) ) |
45 |
44
|
abbii |
⊢ { 𝑧 ∣ ∃ 𝑢 ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴 ) } = { 𝑧 ∣ ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) } ) } |
46 |
|
df-uni |
⊢ ∪ ∪ 𝐴 = { 𝑧 ∣ ∃ 𝑢 ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴 ) } |
47 |
|
df-uni |
⊢ ∪ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) } = { 𝑧 ∣ ∃ 𝑣 ( 𝑧 ∈ 𝑣 ∧ 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) } ) } |
48 |
45 46 47
|
3eqtr4i |
⊢ ∪ ∪ 𝐴 = ∪ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴 ) } |