Step | Hyp | Ref
| Expression |
1 | | rexcom4 3198 |
. . . . . . . . . . 11
⊢
(∃𝑣 ∈
𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔
∃𝑧∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
2 | | r19.41v 3070 |
. . . . . . . . . . . 12
⊢
(∃𝑣 ∈
𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔
(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
3 | 2 | exbii 1764 |
. . . . . . . . . . 11
⊢
(∃𝑧∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔
∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
4 | 1, 3 | bitri 263 |
. . . . . . . . . 10
⊢
(∃𝑣 ∈
𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔
∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
5 | | df-rex 2902 |
. . . . . . . . . . 11
⊢
(∃𝑧 ∈
𝑣 𝑝 = [𝑧] ∼ ↔ ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
6 | 5 | rexbii 3023 |
. . . . . . . . . 10
⊢
(∃𝑣 ∈
𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ↔ ∃𝑣 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
7 | | vex 3176 |
. . . . . . . . . . . 12
⊢ 𝑝 ∈ V |
8 | 7 | elqs 7686 |
. . . . . . . . . . 11
⊢ (𝑝 ∈ (∪ 𝐴
/ ∼ ) ↔
∃𝑧 ∈ ∪ 𝐴𝑝 = [𝑧] ∼ ) |
9 | | df-rex 2902 |
. . . . . . . . . . . 12
⊢
(∃𝑧 ∈
∪ 𝐴𝑝 = [𝑧] ∼ ↔ ∃𝑧(𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ )) |
10 | | eluni2 4376 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ ∪ 𝐴
↔ ∃𝑣 ∈
𝐴 𝑧 ∈ 𝑣) |
11 | 10 | anbi1i 727 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ ∪ 𝐴
∧ 𝑝 = [𝑧] ∼ ) ↔
(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
12 | 11 | exbii 1764 |
. . . . . . . . . . . 12
⊢
(∃𝑧(𝑧 ∈ ∪ 𝐴
∧ 𝑝 = [𝑧] ∼ ) ↔
∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
13 | 9, 12 | bitri 263 |
. . . . . . . . . . 11
⊢
(∃𝑧 ∈
∪ 𝐴𝑝 = [𝑧] ∼ ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
14 | 8, 13 | bitri 263 |
. . . . . . . . . 10
⊢ (𝑝 ∈ (∪ 𝐴
/ ∼ ) ↔
∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
15 | 4, 6, 14 | 3bitr4ri 292 |
. . . . . . . . 9
⊢ (𝑝 ∈ (∪ 𝐴
/ ∼ ) ↔
∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ) |
16 | | prtlem18.1 |
. . . . . . . . . . . 12
⊢ ∼ =
{〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} |
17 | 16 | prtlem19 33181 |
. . . . . . . . . . 11
⊢ (Prt
𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧] ∼ )) |
18 | 17 | ralrimivv 2953 |
. . . . . . . . . 10
⊢ (Prt
𝐴 → ∀𝑣 ∈ 𝐴 ∀𝑧 ∈ 𝑣 𝑣 = [𝑧] ∼ ) |
19 | | 2r19.29 33160 |
. . . . . . . . . . 11
⊢
((∀𝑣 ∈
𝐴 ∀𝑧 ∈ 𝑣 𝑣 = [𝑧] ∼ ∧ ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ) →
∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ )) |
20 | 19 | ex 449 |
. . . . . . . . . 10
⊢
(∀𝑣 ∈
𝐴 ∀𝑧 ∈ 𝑣 𝑣 = [𝑧] ∼ →
(∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ → ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼
))) |
21 | 18, 20 | syl 17 |
. . . . . . . . 9
⊢ (Prt
𝐴 → (∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ → ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼
))) |
22 | 15, 21 | syl5bi 231 |
. . . . . . . 8
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) →
∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼
))) |
23 | | eqtr3 2631 |
. . . . . . . . . 10
⊢ ((𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ) → 𝑣 = 𝑝) |
24 | 23 | reximi 2994 |
. . . . . . . . 9
⊢
(∃𝑧 ∈
𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ) →
∃𝑧 ∈ 𝑣 𝑣 = 𝑝) |
25 | 24 | reximi 2994 |
. . . . . . . 8
⊢
(∃𝑣 ∈
𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ) →
∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑣 = 𝑝) |
26 | 22, 25 | syl6 34 |
. . . . . . 7
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) →
∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑣 = 𝑝)) |
27 | | df-rex 2902 |
. . . . . . . . . 10
⊢
(∃𝑧 ∈
𝑣 𝑣 = 𝑝 ↔ ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝)) |
28 | | 19.41v 1901 |
. . . . . . . . . 10
⊢
(∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝) ↔ (∃𝑧 𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝)) |
29 | 27, 28 | bitri 263 |
. . . . . . . . 9
⊢
(∃𝑧 ∈
𝑣 𝑣 = 𝑝 ↔ (∃𝑧 𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝)) |
30 | 29 | simprbi 479 |
. . . . . . . 8
⊢
(∃𝑧 ∈
𝑣 𝑣 = 𝑝 → 𝑣 = 𝑝) |
31 | 30 | reximi 2994 |
. . . . . . 7
⊢
(∃𝑣 ∈
𝐴 ∃𝑧 ∈ 𝑣 𝑣 = 𝑝 → ∃𝑣 ∈ 𝐴 𝑣 = 𝑝) |
32 | 26, 31 | syl6 34 |
. . . . . 6
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) →
∃𝑣 ∈ 𝐴 𝑣 = 𝑝)) |
33 | | risset 3044 |
. . . . . 6
⊢ (𝑝 ∈ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑣 = 𝑝) |
34 | 32, 33 | syl6ibr 241 |
. . . . 5
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) → 𝑝 ∈ 𝐴)) |
35 | 16 | prtlem400 33173 |
. . . . . 6
⊢ ¬
∅ ∈ (∪ 𝐴 / ∼ ) |
36 | | nelelne 2880 |
. . . . . 6
⊢ (¬
∅ ∈ (∪ 𝐴 / ∼ ) → (𝑝 ∈ (∪ 𝐴
/ ∼ ) → 𝑝 ≠ ∅)) |
37 | 35, 36 | mp1i 13 |
. . . . 5
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) → 𝑝 ≠ ∅)) |
38 | 34, 37 | jcad 554 |
. . . 4
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) → (𝑝 ∈ 𝐴 ∧ 𝑝 ≠ ∅))) |
39 | | eldifsn 4260 |
. . . 4
⊢ (𝑝 ∈ (𝐴 ∖ {∅}) ↔ (𝑝 ∈ 𝐴 ∧ 𝑝 ≠ ∅)) |
40 | 38, 39 | syl6ibr 241 |
. . 3
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) → 𝑝 ∈ (𝐴 ∖ {∅}))) |
41 | | neldifsn 4262 |
. . . . . . 7
⊢ ¬
∅ ∈ (𝐴 ∖
{∅}) |
42 | | n0el 33164 |
. . . . . . 7
⊢ (¬
∅ ∈ (𝐴 ∖
{∅}) ↔ ∀𝑝
∈ (𝐴 ∖
{∅})∃𝑧 𝑧 ∈ 𝑝) |
43 | 41, 42 | mpbi 219 |
. . . . . 6
⊢
∀𝑝 ∈
(𝐴 ∖
{∅})∃𝑧 𝑧 ∈ 𝑝 |
44 | 43 | rspec 2915 |
. . . . 5
⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → ∃𝑧 𝑧 ∈ 𝑝) |
45 | | eldifi 3694 |
. . . . 5
⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝 ∈ 𝐴) |
46 | 44, 45 | jca 553 |
. . . 4
⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → (∃𝑧 𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴)) |
47 | 16 | prtlem19 33181 |
. . . . . . . . 9
⊢ (Prt
𝐴 → ((𝑝 ∈ 𝐴 ∧ 𝑧 ∈ 𝑝) → 𝑝 = [𝑧] ∼ )) |
48 | 47 | ancomsd 469 |
. . . . . . . 8
⊢ (Prt
𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑝 = [𝑧] ∼ )) |
49 | | elunii 4377 |
. . . . . . . 8
⊢ ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) |
50 | 48, 49 | jca2r 33155 |
. . . . . . 7
⊢ (Prt
𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → (𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼
))) |
51 | | prtlem11 33169 |
. . . . . . . . 9
⊢ (𝑝 ∈ V → (𝑧 ∈ ∪ 𝐴
→ (𝑝 = [𝑧] ∼ → 𝑝 ∈ (∪ 𝐴
/ ∼
)))) |
52 | 7, 51 | ax-mp 5 |
. . . . . . . 8
⊢ (𝑧 ∈ ∪ 𝐴
→ (𝑝 = [𝑧] ∼ → 𝑝 ∈ (∪ 𝐴
/ ∼
))) |
53 | 52 | imp 444 |
. . . . . . 7
⊢ ((𝑧 ∈ ∪ 𝐴
∧ 𝑝 = [𝑧] ∼ ) → 𝑝 ∈ (∪ 𝐴
/ ∼ )) |
54 | 50, 53 | syl6 34 |
. . . . . 6
⊢ (Prt
𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ (∪ 𝐴 / ∼
))) |
55 | 54 | eximdv 1833 |
. . . . 5
⊢ (Prt
𝐴 → (∃𝑧(𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → ∃𝑧 𝑝 ∈ (∪ 𝐴 / ∼
))) |
56 | | 19.41v 1901 |
. . . . 5
⊢
(∃𝑧(𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) ↔ (∃𝑧 𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴)) |
57 | | 19.9v 1883 |
. . . . 5
⊢
(∃𝑧 𝑝 ∈ (∪ 𝐴
/ ∼ ) ↔ 𝑝 ∈ (∪ 𝐴
/ ∼ )) |
58 | 55, 56, 57 | 3imtr3g 283 |
. . . 4
⊢ (Prt
𝐴 → ((∃𝑧 𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ (∪ 𝐴 / ∼
))) |
59 | 46, 58 | syl5 33 |
. . 3
⊢ (Prt
𝐴 → (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝 ∈ (∪ 𝐴
/ ∼
))) |
60 | 40, 59 | impbid 201 |
. 2
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) ↔ 𝑝 ∈ (𝐴 ∖ {∅}))) |
61 | 60 | eqrdv 2608 |
1
⊢ (Prt
𝐴 → (∪ 𝐴
/ ∼ ) = (𝐴 ∖
{∅})) |