Proof of Theorem umgrislfupgrlem
Step | Hyp | Ref
| Expression |
1 | | 2pos 12076 |
. . . 4
⊢ 0 <
2 |
2 | | simprl 768 |
. . . . . . . . 9
⊢ ((0 <
2 ∧ (𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥))) →
𝑥 ∈ 𝒫 𝑉) |
3 | | fveq2 6771 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = ∅ →
(♯‘𝑥) =
(♯‘∅)) |
4 | | hash0 14080 |
. . . . . . . . . . . . . . 15
⊢
(♯‘∅) = 0 |
5 | 3, 4 | eqtrdi 2796 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = ∅ →
(♯‘𝑥) =
0) |
6 | 5 | breq2d 5091 |
. . . . . . . . . . . . 13
⊢ (𝑥 = ∅ → (2 ≤
(♯‘𝑥) ↔ 2
≤ 0)) |
7 | | 2re 12047 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℝ |
8 | | 0re 10978 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
ℝ |
9 | 7, 8 | lenlti 11095 |
. . . . . . . . . . . . . 14
⊢ (2 ≤ 0
↔ ¬ 0 < 2) |
10 | | pm2.21 123 |
. . . . . . . . . . . . . 14
⊢ (¬ 0
< 2 → (0 < 2 → 𝑥 ≠ ∅)) |
11 | 9, 10 | sylbi 216 |
. . . . . . . . . . . . 13
⊢ (2 ≤ 0
→ (0 < 2 → 𝑥
≠ ∅)) |
12 | 6, 11 | syl6bi 252 |
. . . . . . . . . . . 12
⊢ (𝑥 = ∅ → (2 ≤
(♯‘𝑥) → (0
< 2 → 𝑥 ≠
∅))) |
13 | 12 | adantld 491 |
. . . . . . . . . . 11
⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝒫 𝑉 ∧ 2 ≤
(♯‘𝑥)) →
(0 < 2 → 𝑥 ≠
∅))) |
14 | 13 | impcomd 412 |
. . . . . . . . . 10
⊢ (𝑥 = ∅ → ((0 < 2
∧ (𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥))) →
𝑥 ≠
∅)) |
15 | | ax-1 6 |
. . . . . . . . . 10
⊢ (𝑥 ≠ ∅ → ((0 < 2
∧ (𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥))) →
𝑥 ≠
∅)) |
16 | 14, 15 | pm2.61ine 3030 |
. . . . . . . . 9
⊢ ((0 <
2 ∧ (𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥))) →
𝑥 ≠
∅) |
17 | | eldifsn 4726 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ↔
(𝑥 ∈ 𝒫 𝑉 ∧ 𝑥 ≠ ∅)) |
18 | 2, 16, 17 | sylanbrc 583 |
. . . . . . . 8
⊢ ((0 <
2 ∧ (𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥))) →
𝑥 ∈ (𝒫 𝑉 ∖
{∅})) |
19 | | simprr 770 |
. . . . . . . 8
⊢ ((0 <
2 ∧ (𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥))) →
2 ≤ (♯‘𝑥)) |
20 | 18, 19 | jca 512 |
. . . . . . 7
⊢ ((0 <
2 ∧ (𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥))) →
(𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 2 ≤
(♯‘𝑥))) |
21 | 20 | ex 413 |
. . . . . 6
⊢ (0 < 2
→ ((𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥)) →
(𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 2 ≤
(♯‘𝑥)))) |
22 | | eldifi 4066 |
. . . . . . 7
⊢ (𝑥 ∈ (𝒫 𝑉 ∖ {∅}) → 𝑥 ∈ 𝒫 𝑉) |
23 | 22 | anim1i 615 |
. . . . . 6
⊢ ((𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 2 ≤
(♯‘𝑥)) →
(𝑥 ∈ 𝒫 𝑉 ∧ 2 ≤
(♯‘𝑥))) |
24 | 21, 23 | impbid1 224 |
. . . . 5
⊢ (0 < 2
→ ((𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥)) ↔
(𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 2 ≤
(♯‘𝑥)))) |
25 | 24 | rabbidva2 3409 |
. . . 4
⊢ (0 < 2
→ {𝑥 ∈ 𝒫
𝑉 ∣ 2 ≤
(♯‘𝑥)} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ 2
≤ (♯‘𝑥)}) |
26 | 1, 25 | ax-mp 5 |
. . 3
⊢ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤
(♯‘𝑥)} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ 2
≤ (♯‘𝑥)} |
27 | 26 | ineq2i 4149 |
. 2
⊢ ({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑥) ≤ 2}
∩ {𝑥 ∈ 𝒫
𝑉 ∣ 2 ≤
(♯‘𝑥)}) =
({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑥) ≤ 2}
∩ {𝑥 ∈ (𝒫
𝑉 ∖ {∅})
∣ 2 ≤ (♯‘𝑥)}) |
28 | | inrab 4246 |
. 2
⊢ ({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑥) ≤ 2}
∩ {𝑥 ∈ (𝒫
𝑉 ∖ {∅})
∣ 2 ≤ (♯‘𝑥)}) = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
((♯‘𝑥) ≤ 2
∧ 2 ≤ (♯‘𝑥))} |
29 | | hashxnn0 14051 |
. . . . . . 7
⊢ (𝑥 ∈ V →
(♯‘𝑥) ∈
ℕ0*) |
30 | 29 | elv 3437 |
. . . . . 6
⊢
(♯‘𝑥)
∈ ℕ0* |
31 | | xnn0xr 12310 |
. . . . . 6
⊢
((♯‘𝑥)
∈ ℕ0* → (♯‘𝑥) ∈
ℝ*) |
32 | 30, 31 | ax-mp 5 |
. . . . 5
⊢
(♯‘𝑥)
∈ ℝ* |
33 | 7 | rexri 11034 |
. . . . 5
⊢ 2 ∈
ℝ* |
34 | | xrletri3 12887 |
. . . . 5
⊢
(((♯‘𝑥)
∈ ℝ* ∧ 2 ∈ ℝ*) →
((♯‘𝑥) = 2
↔ ((♯‘𝑥)
≤ 2 ∧ 2 ≤ (♯‘𝑥)))) |
35 | 32, 33, 34 | mp2an 689 |
. . . 4
⊢
((♯‘𝑥) =
2 ↔ ((♯‘𝑥)
≤ 2 ∧ 2 ≤ (♯‘𝑥))) |
36 | 35 | bicomi 223 |
. . 3
⊢
(((♯‘𝑥)
≤ 2 ∧ 2 ≤ (♯‘𝑥)) ↔ (♯‘𝑥) = 2) |
37 | 36 | rabbii 3406 |
. 2
⊢ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
((♯‘𝑥) ≤ 2
∧ 2 ≤ (♯‘𝑥))} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑥) =
2} |
38 | 27, 28, 37 | 3eqtri 2772 |
1
⊢ ({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑥) ≤ 2}
∩ {𝑥 ∈ 𝒫
𝑉 ∣ 2 ≤
(♯‘𝑥)}) =
{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑥) =
2} |