| Step | Hyp | Ref
| Expression |
| 1 | | rexcom4 3288 |
. . . . . . . . . . 11
⊢
(∃𝑣 ∈
𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔
∃𝑧∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
| 2 | | r19.41v 3189 |
. . . . . . . . . . . 12
⊢
(∃𝑣 ∈
𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔
(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
| 3 | 2 | exbii 1848 |
. . . . . . . . . . 11
⊢
(∃𝑧∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔
∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
| 4 | 1, 3 | bitri 275 |
. . . . . . . . . 10
⊢
(∃𝑣 ∈
𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ ) ↔
∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
| 5 | | df-rex 3071 |
. . . . . . . . . . 11
⊢
(∃𝑧 ∈
𝑣 𝑝 = [𝑧] ∼ ↔ ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
| 6 | 5 | rexbii 3094 |
. . . . . . . . . 10
⊢
(∃𝑣 ∈
𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ↔ ∃𝑣 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
| 7 | | vex 3484 |
. . . . . . . . . . . 12
⊢ 𝑝 ∈ V |
| 8 | 7 | elqs 8809 |
. . . . . . . . . . 11
⊢ (𝑝 ∈ (∪ 𝐴
/ ∼ ) ↔
∃𝑧 ∈ ∪ 𝐴𝑝 = [𝑧] ∼ ) |
| 9 | | df-rex 3071 |
. . . . . . . . . . . 12
⊢
(∃𝑧 ∈
∪ 𝐴𝑝 = [𝑧] ∼ ↔ ∃𝑧(𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼ )) |
| 10 | | eluni2 4911 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ ∪ 𝐴
↔ ∃𝑣 ∈
𝐴 𝑧 ∈ 𝑣) |
| 11 | 10 | anbi1i 624 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ ∪ 𝐴
∧ 𝑝 = [𝑧] ∼ ) ↔
(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
| 12 | 11 | exbii 1848 |
. . . . . . . . . . . 12
⊢
(∃𝑧(𝑧 ∈ ∪ 𝐴
∧ 𝑝 = [𝑧] ∼ ) ↔
∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
| 13 | 9, 12 | bitri 275 |
. . . . . . . . . . 11
⊢
(∃𝑧 ∈
∪ 𝐴𝑝 = [𝑧] ∼ ↔ ∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
| 14 | 8, 13 | bitri 275 |
. . . . . . . . . 10
⊢ (𝑝 ∈ (∪ 𝐴
/ ∼ ) ↔
∃𝑧(∃𝑣 ∈ 𝐴 𝑧 ∈ 𝑣 ∧ 𝑝 = [𝑧] ∼ )) |
| 15 | 4, 6, 14 | 3bitr4ri 304 |
. . . . . . . . 9
⊢ (𝑝 ∈ (∪ 𝐴
/ ∼ ) ↔
∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ) |
| 16 | | prtlem18.1 |
. . . . . . . . . . . 12
⊢ ∼ =
{〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} |
| 17 | 16 | prtlem19 38879 |
. . . . . . . . . . 11
⊢ (Prt
𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧] ∼ )) |
| 18 | 17 | ralrimivv 3200 |
. . . . . . . . . 10
⊢ (Prt
𝐴 → ∀𝑣 ∈ 𝐴 ∀𝑧 ∈ 𝑣 𝑣 = [𝑧] ∼ ) |
| 19 | | 2r19.29 3139 |
. . . . . . . . . . 11
⊢
((∀𝑣 ∈
𝐴 ∀𝑧 ∈ 𝑣 𝑣 = [𝑧] ∼ ∧ ∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑝 = [𝑧] ∼ ) →
∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ )) |
| 20 | 19 | ex 412 |
. . . . . . . . . 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 | biimtrid 242 |
. . . . . . . 8
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) →
∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼
))) |
| 23 | | eqtr3 2763 |
. . . . . . . . . 10
⊢ ((𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ) → 𝑣 = 𝑝) |
| 24 | 23 | reximi 3084 |
. . . . . . . . 9
⊢
(∃𝑧 ∈
𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ) →
∃𝑧 ∈ 𝑣 𝑣 = 𝑝) |
| 25 | 24 | reximi 3084 |
. . . . . . . 8
⊢
(∃𝑣 ∈
𝐴 ∃𝑧 ∈ 𝑣 (𝑣 = [𝑧] ∼ ∧ 𝑝 = [𝑧] ∼ ) →
∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑣 = 𝑝) |
| 26 | 22, 25 | syl6 35 |
. . . . . . 7
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) →
∃𝑣 ∈ 𝐴 ∃𝑧 ∈ 𝑣 𝑣 = 𝑝)) |
| 27 | | df-rex 3071 |
. . . . . . . . . 10
⊢
(∃𝑧 ∈
𝑣 𝑣 = 𝑝 ↔ ∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝)) |
| 28 | | 19.41v 1949 |
. . . . . . . . . 10
⊢
(∃𝑧(𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝) ↔ (∃𝑧 𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝)) |
| 29 | 27, 28 | bitri 275 |
. . . . . . . . 9
⊢
(∃𝑧 ∈
𝑣 𝑣 = 𝑝 ↔ (∃𝑧 𝑧 ∈ 𝑣 ∧ 𝑣 = 𝑝)) |
| 30 | 29 | simprbi 496 |
. . . . . . . 8
⊢
(∃𝑧 ∈
𝑣 𝑣 = 𝑝 → 𝑣 = 𝑝) |
| 31 | 30 | reximi 3084 |
. . . . . . 7
⊢
(∃𝑣 ∈
𝐴 ∃𝑧 ∈ 𝑣 𝑣 = 𝑝 → ∃𝑣 ∈ 𝐴 𝑣 = 𝑝) |
| 32 | 26, 31 | syl6 35 |
. . . . . 6
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) →
∃𝑣 ∈ 𝐴 𝑣 = 𝑝)) |
| 33 | | risset 3233 |
. . . . . 6
⊢ (𝑝 ∈ 𝐴 ↔ ∃𝑣 ∈ 𝐴 𝑣 = 𝑝) |
| 34 | 32, 33 | imbitrrdi 252 |
. . . . 5
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) → 𝑝 ∈ 𝐴)) |
| 35 | 16 | prtlem400 38871 |
. . . . . 6
⊢ ¬
∅ ∈ (∪ 𝐴 / ∼ ) |
| 36 | | nelelne 3041 |
. . . . . 6
⊢ (¬
∅ ∈ (∪ 𝐴 / ∼ ) → (𝑝 ∈ (∪ 𝐴
/ ∼ ) → 𝑝 ≠ ∅)) |
| 37 | 35, 36 | mp1i 13 |
. . . . 5
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) → 𝑝 ≠ ∅)) |
| 38 | 34, 37 | jcad 512 |
. . . 4
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) → (𝑝 ∈ 𝐴 ∧ 𝑝 ≠ ∅))) |
| 39 | | eldifsn 4786 |
. . . 4
⊢ (𝑝 ∈ (𝐴 ∖ {∅}) ↔ (𝑝 ∈ 𝐴 ∧ 𝑝 ≠ ∅)) |
| 40 | 38, 39 | imbitrrdi 252 |
. . 3
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) → 𝑝 ∈ (𝐴 ∖ {∅}))) |
| 41 | | neldifsn 4792 |
. . . . . . 7
⊢ ¬
∅ ∈ (𝐴 ∖
{∅}) |
| 42 | | n0el 4364 |
. . . . . . 7
⊢ (¬
∅ ∈ (𝐴 ∖
{∅}) ↔ ∀𝑝
∈ (𝐴 ∖
{∅})∃𝑧 𝑧 ∈ 𝑝) |
| 43 | 41, 42 | mpbi 230 |
. . . . . 6
⊢
∀𝑝 ∈
(𝐴 ∖
{∅})∃𝑧 𝑧 ∈ 𝑝 |
| 44 | 43 | rspec 3250 |
. . . . 5
⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → ∃𝑧 𝑧 ∈ 𝑝) |
| 45 | | eldifi 4131 |
. . . . 5
⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝 ∈ 𝐴) |
| 46 | 44, 45 | jca 511 |
. . . 4
⊢ (𝑝 ∈ (𝐴 ∖ {∅}) → (∃𝑧 𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴)) |
| 47 | 16 | prtlem19 38879 |
. . . . . . . . 9
⊢ (Prt
𝐴 → ((𝑝 ∈ 𝐴 ∧ 𝑧 ∈ 𝑝) → 𝑝 = [𝑧] ∼ )) |
| 48 | 47 | ancomsd 465 |
. . . . . . . 8
⊢ (Prt
𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑝 = [𝑧] ∼ )) |
| 49 | | elunii 4912 |
. . . . . . . 8
⊢ ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) |
| 50 | 48, 49 | jca2r 38856 |
. . . . . . 7
⊢ (Prt
𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → (𝑧 ∈ ∪ 𝐴 ∧ 𝑝 = [𝑧] ∼
))) |
| 51 | | prtlem11 38867 |
. . . . . . . . 9
⊢ (𝑝 ∈ V → (𝑧 ∈ ∪ 𝐴
→ (𝑝 = [𝑧] ∼ → 𝑝 ∈ (∪ 𝐴
/ ∼
)))) |
| 52 | 51 | elv 3485 |
. . . . . . . 8
⊢ (𝑧 ∈ ∪ 𝐴
→ (𝑝 = [𝑧] ∼ → 𝑝 ∈ (∪ 𝐴
/ ∼
))) |
| 53 | 52 | imp 406 |
. . . . . . 7
⊢ ((𝑧 ∈ ∪ 𝐴
∧ 𝑝 = [𝑧] ∼ ) → 𝑝 ∈ (∪ 𝐴
/ ∼ )) |
| 54 | 50, 53 | syl6 35 |
. . . . . 6
⊢ (Prt
𝐴 → ((𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ (∪ 𝐴 / ∼
))) |
| 55 | 54 | eximdv 1917 |
. . . . 5
⊢ (Prt
𝐴 → (∃𝑧(𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → ∃𝑧 𝑝 ∈ (∪ 𝐴 / ∼
))) |
| 56 | | 19.41v 1949 |
. . . . 5
⊢
(∃𝑧(𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) ↔ (∃𝑧 𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴)) |
| 57 | | 19.9v 1983 |
. . . . 5
⊢
(∃𝑧 𝑝 ∈ (∪ 𝐴
/ ∼ ) ↔ 𝑝 ∈ (∪ 𝐴
/ ∼ )) |
| 58 | 55, 56, 57 | 3imtr3g 295 |
. . . 4
⊢ (Prt
𝐴 → ((∃𝑧 𝑧 ∈ 𝑝 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ (∪ 𝐴 / ∼
))) |
| 59 | 46, 58 | syl5 34 |
. . 3
⊢ (Prt
𝐴 → (𝑝 ∈ (𝐴 ∖ {∅}) → 𝑝 ∈ (∪ 𝐴
/ ∼
))) |
| 60 | 40, 59 | impbid 212 |
. 2
⊢ (Prt
𝐴 → (𝑝 ∈ (∪ 𝐴
/ ∼ ) ↔ 𝑝 ∈ (𝐴 ∖ {∅}))) |
| 61 | 60 | eqrdv 2735 |
1
⊢ (Prt
𝐴 → (∪ 𝐴
/ ∼ ) = (𝐴 ∖
{∅})) |