Proof of Theorem umgrislfupgrlem
Step | Hyp | Ref
| Expression |
1 | | 2pos 11275 |
. . . 4
⊢ 0 <
2 |
2 | | simprl 811 |
. . . . . . . . 9
⊢ ((0 <
2 ∧ (𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥))) →
𝑥 ∈ 𝒫 𝑉) |
3 | | fveq2 6340 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = ∅ →
(♯‘𝑥) =
(♯‘∅)) |
4 | | hash0 13321 |
. . . . . . . . . . . . . . . 16
⊢
(♯‘∅) = 0 |
5 | 3, 4 | syl6eq 2798 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = ∅ →
(♯‘𝑥) =
0) |
6 | 5 | breq2d 4804 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = ∅ → (2 ≤
(♯‘𝑥) ↔ 2
≤ 0)) |
7 | | 2re 11253 |
. . . . . . . . . . . . . . . 16
⊢ 2 ∈
ℝ |
8 | | 0re 10203 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
ℝ |
9 | 7, 8 | lenlti 10320 |
. . . . . . . . . . . . . . 15
⊢ (2 ≤ 0
↔ ¬ 0 < 2) |
10 | | pm2.21 120 |
. . . . . . . . . . . . . . 15
⊢ (¬ 0
< 2 → (0 < 2 → 𝑥 ≠ ∅)) |
11 | 9, 10 | sylbi 207 |
. . . . . . . . . . . . . 14
⊢ (2 ≤ 0
→ (0 < 2 → 𝑥
≠ ∅)) |
12 | 6, 11 | syl6bi 243 |
. . . . . . . . . . . . 13
⊢ (𝑥 = ∅ → (2 ≤
(♯‘𝑥) → (0
< 2 → 𝑥 ≠
∅))) |
13 | 12 | adantld 484 |
. . . . . . . . . . . 12
⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝒫 𝑉 ∧ 2 ≤
(♯‘𝑥)) →
(0 < 2 → 𝑥 ≠
∅))) |
14 | 13 | com23 86 |
. . . . . . . . . . 11
⊢ (𝑥 = ∅ → (0 < 2
→ ((𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥)) →
𝑥 ≠
∅))) |
15 | 14 | impd 446 |
. . . . . . . . . 10
⊢ (𝑥 = ∅ → ((0 < 2
∧ (𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥))) →
𝑥 ≠
∅)) |
16 | | ax-1 6 |
. . . . . . . . . 10
⊢ (𝑥 ≠ ∅ → ((0 < 2
∧ (𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥))) →
𝑥 ≠
∅)) |
17 | 15, 16 | pm2.61ine 3003 |
. . . . . . . . 9
⊢ ((0 <
2 ∧ (𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥))) →
𝑥 ≠
∅) |
18 | | eldifsn 4450 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ↔
(𝑥 ∈ 𝒫 𝑉 ∧ 𝑥 ≠ ∅)) |
19 | 2, 17, 18 | sylanbrc 701 |
. . . . . . . 8
⊢ ((0 <
2 ∧ (𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥))) →
𝑥 ∈ (𝒫 𝑉 ∖
{∅})) |
20 | | simprr 813 |
. . . . . . . 8
⊢ ((0 <
2 ∧ (𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥))) →
2 ≤ (♯‘𝑥)) |
21 | 19, 20 | jca 555 |
. . . . . . 7
⊢ ((0 <
2 ∧ (𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥))) →
(𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 2 ≤
(♯‘𝑥))) |
22 | 21 | ex 449 |
. . . . . 6
⊢ (0 < 2
→ ((𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥)) →
(𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 2 ≤
(♯‘𝑥)))) |
23 | | eldifi 3863 |
. . . . . . 7
⊢ (𝑥 ∈ (𝒫 𝑉 ∖ {∅}) → 𝑥 ∈ 𝒫 𝑉) |
24 | 23 | anim1i 593 |
. . . . . 6
⊢ ((𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 2 ≤
(♯‘𝑥)) →
(𝑥 ∈ 𝒫 𝑉 ∧ 2 ≤
(♯‘𝑥))) |
25 | 22, 24 | impbid1 215 |
. . . . 5
⊢ (0 < 2
→ ((𝑥 ∈ 𝒫
𝑉 ∧ 2 ≤
(♯‘𝑥)) ↔
(𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 2 ≤
(♯‘𝑥)))) |
26 | 25 | rabbidva2 3314 |
. . . 4
⊢ (0 < 2
→ {𝑥 ∈ 𝒫
𝑉 ∣ 2 ≤
(♯‘𝑥)} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ 2
≤ (♯‘𝑥)}) |
27 | 1, 26 | ax-mp 5 |
. . 3
⊢ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤
(♯‘𝑥)} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ 2
≤ (♯‘𝑥)} |
28 | 27 | ineq2i 3942 |
. 2
⊢ ({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑥) ≤ 2}
∩ {𝑥 ∈ 𝒫
𝑉 ∣ 2 ≤
(♯‘𝑥)}) =
({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑥) ≤ 2}
∩ {𝑥 ∈ (𝒫
𝑉 ∖ {∅})
∣ 2 ≤ (♯‘𝑥)}) |
29 | | inrab 4030 |
. 2
⊢ ({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑥) ≤ 2}
∩ {𝑥 ∈ (𝒫
𝑉 ∖ {∅})
∣ 2 ≤ (♯‘𝑥)}) = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
((♯‘𝑥) ≤ 2
∧ 2 ≤ (♯‘𝑥))} |
30 | | vex 3331 |
. . . . . . 7
⊢ 𝑥 ∈ V |
31 | | hashxnn0 13292 |
. . . . . . 7
⊢ (𝑥 ∈ V →
(♯‘𝑥) ∈
ℕ0*) |
32 | 30, 31 | ax-mp 5 |
. . . . . 6
⊢
(♯‘𝑥)
∈ ℕ0* |
33 | | xnn0xr 11531 |
. . . . . 6
⊢
((♯‘𝑥)
∈ ℕ0* → (♯‘𝑥) ∈
ℝ*) |
34 | 32, 33 | ax-mp 5 |
. . . . 5
⊢
(♯‘𝑥)
∈ ℝ* |
35 | 7 | rexri 10260 |
. . . . 5
⊢ 2 ∈
ℝ* |
36 | | xrletri3 12149 |
. . . . 5
⊢
(((♯‘𝑥)
∈ ℝ* ∧ 2 ∈ ℝ*) →
((♯‘𝑥) = 2
↔ ((♯‘𝑥)
≤ 2 ∧ 2 ≤ (♯‘𝑥)))) |
37 | 34, 35, 36 | mp2an 710 |
. . . 4
⊢
((♯‘𝑥) =
2 ↔ ((♯‘𝑥)
≤ 2 ∧ 2 ≤ (♯‘𝑥))) |
38 | 37 | bicomi 214 |
. . 3
⊢
(((♯‘𝑥)
≤ 2 ∧ 2 ≤ (♯‘𝑥)) ↔ (♯‘𝑥) = 2) |
39 | 38 | rabbii 3313 |
. 2
⊢ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
((♯‘𝑥) ≤ 2
∧ 2 ≤ (♯‘𝑥))} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑥) =
2} |
40 | 28, 29, 39 | 3eqtri 2774 |
1
⊢ ({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑥) ≤ 2}
∩ {𝑥 ∈ 𝒫
𝑉 ∣ 2 ≤
(♯‘𝑥)}) =
{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑥) =
2} |