Proof of Theorem hashge2el2dif
| Step | Hyp | Ref
| Expression |
| 1 | | fveq2 6906 |
. . . . . . 7
⊢ (𝐷 = {𝑥} → (♯‘𝐷) = (♯‘{𝑥})) |
| 2 | | hashsng 14408 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐷 → (♯‘{𝑥}) = 1) |
| 3 | 1, 2 | sylan9eqr 2799 |
. . . . . 6
⊢ ((𝑥 ∈ 𝐷 ∧ 𝐷 = {𝑥}) → (♯‘𝐷) = 1) |
| 4 | 3 | ralimiaa 3082 |
. . . . 5
⊢
(∀𝑥 ∈
𝐷 𝐷 = {𝑥} → ∀𝑥 ∈ 𝐷 (♯‘𝐷) = 1) |
| 5 | | 0re 11263 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
ℝ |
| 6 | | 1re 11261 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℝ |
| 7 | 5, 6 | readdcli 11276 |
. . . . . . . . . . . . 13
⊢ (0 + 1)
∈ ℝ |
| 8 | 7 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝐷 ∈ Fin ∧ (𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷))) → (0 + 1) ∈
ℝ) |
| 9 | | 2re 12340 |
. . . . . . . . . . . . 13
⊢ 2 ∈
ℝ |
| 10 | 9 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝐷 ∈ Fin ∧ (𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷))) → 2 ∈
ℝ) |
| 11 | | hashcl 14395 |
. . . . . . . . . . . . . 14
⊢ (𝐷 ∈ Fin →
(♯‘𝐷) ∈
ℕ0) |
| 12 | 11 | nn0red 12588 |
. . . . . . . . . . . . 13
⊢ (𝐷 ∈ Fin →
(♯‘𝐷) ∈
ℝ) |
| 13 | 12 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝐷 ∈ Fin ∧ (𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷))) → (♯‘𝐷) ∈
ℝ) |
| 14 | 8, 10, 13 | 3jca 1129 |
. . . . . . . . . . 11
⊢ ((𝐷 ∈ Fin ∧ (𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷))) → ((0 + 1) ∈
ℝ ∧ 2 ∈ ℝ ∧ (♯‘𝐷) ∈ ℝ)) |
| 15 | | 0p1e1 12388 |
. . . . . . . . . . . . . . 15
⊢ (0 + 1) =
1 |
| 16 | | 1lt2 12437 |
. . . . . . . . . . . . . . 15
⊢ 1 <
2 |
| 17 | 15, 16 | eqbrtri 5164 |
. . . . . . . . . . . . . 14
⊢ (0 + 1)
< 2 |
| 18 | 17 | jctl 523 |
. . . . . . . . . . . . 13
⊢ (2 ≤
(♯‘𝐷) →
((0 + 1) < 2 ∧ 2 ≤ (♯‘𝐷))) |
| 19 | 18 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → ((0 + 1) < 2 ∧
2 ≤ (♯‘𝐷))) |
| 20 | 19 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝐷 ∈ Fin ∧ (𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷))) → ((0 + 1) < 2 ∧
2 ≤ (♯‘𝐷))) |
| 21 | | ltleletr 11354 |
. . . . . . . . . . 11
⊢ (((0 + 1)
∈ ℝ ∧ 2 ∈ ℝ ∧ (♯‘𝐷) ∈ ℝ) → (((0 + 1) < 2
∧ 2 ≤ (♯‘𝐷)) → (0 + 1) ≤ (♯‘𝐷))) |
| 22 | 14, 20, 21 | sylc 65 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ Fin ∧ (𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷))) → (0 + 1) ≤
(♯‘𝐷)) |
| 23 | 11 | nn0zd 12639 |
. . . . . . . . . . . . 13
⊢ (𝐷 ∈ Fin →
(♯‘𝐷) ∈
ℤ) |
| 24 | | 0z 12624 |
. . . . . . . . . . . . 13
⊢ 0 ∈
ℤ |
| 25 | 23, 24 | jctil 519 |
. . . . . . . . . . . 12
⊢ (𝐷 ∈ Fin → (0 ∈
ℤ ∧ (♯‘𝐷) ∈ ℤ)) |
| 26 | 25 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝐷 ∈ Fin ∧ (𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷))) → (0 ∈ ℤ
∧ (♯‘𝐷)
∈ ℤ)) |
| 27 | | zltp1le 12667 |
. . . . . . . . . . 11
⊢ ((0
∈ ℤ ∧ (♯‘𝐷) ∈ ℤ) → (0 <
(♯‘𝐷) ↔ (0
+ 1) ≤ (♯‘𝐷))) |
| 28 | 26, 27 | syl 17 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ Fin ∧ (𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷))) → (0 <
(♯‘𝐷) ↔ (0
+ 1) ≤ (♯‘𝐷))) |
| 29 | 22, 28 | mpbird 257 |
. . . . . . . . 9
⊢ ((𝐷 ∈ Fin ∧ (𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷))) → 0 <
(♯‘𝐷)) |
| 30 | | 0ltpnf 13164 |
. . . . . . . . . 10
⊢ 0 <
+∞ |
| 31 | | simpl 482 |
. . . . . . . . . . . . 13
⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → 𝐷 ∈ 𝑉) |
| 32 | 31 | anim2i 617 |
. . . . . . . . . . . 12
⊢ ((¬
𝐷 ∈ Fin ∧ (𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷))) → (¬ 𝐷 ∈ Fin ∧ 𝐷 ∈ 𝑉)) |
| 33 | 32 | ancomd 461 |
. . . . . . . . . . 11
⊢ ((¬
𝐷 ∈ Fin ∧ (𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷))) → (𝐷 ∈ 𝑉 ∧ ¬ 𝐷 ∈ Fin)) |
| 34 | | hashinf 14374 |
. . . . . . . . . . 11
⊢ ((𝐷 ∈ 𝑉 ∧ ¬ 𝐷 ∈ Fin) → (♯‘𝐷) = +∞) |
| 35 | 33, 34 | syl 17 |
. . . . . . . . . 10
⊢ ((¬
𝐷 ∈ Fin ∧ (𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷))) → (♯‘𝐷) = +∞) |
| 36 | 30, 35 | breqtrrid 5181 |
. . . . . . . . 9
⊢ ((¬
𝐷 ∈ Fin ∧ (𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷))) → 0 <
(♯‘𝐷)) |
| 37 | 29, 36 | pm2.61ian 812 |
. . . . . . . 8
⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → 0 <
(♯‘𝐷)) |
| 38 | | hashgt0n0 14404 |
. . . . . . . 8
⊢ ((𝐷 ∈ 𝑉 ∧ 0 < (♯‘𝐷)) → 𝐷 ≠ ∅) |
| 39 | 37, 38 | syldan 591 |
. . . . . . 7
⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → 𝐷 ≠ ∅) |
| 40 | | rspn0 4356 |
. . . . . . 7
⊢ (𝐷 ≠ ∅ →
(∀𝑥 ∈ 𝐷 (♯‘𝐷) = 1 →
(♯‘𝐷) =
1)) |
| 41 | 39, 40 | syl 17 |
. . . . . 6
⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → (∀𝑥 ∈ 𝐷 (♯‘𝐷) = 1 → (♯‘𝐷) = 1)) |
| 42 | | breq2 5147 |
. . . . . . . . 9
⊢
((♯‘𝐷) =
1 → (2 ≤ (♯‘𝐷) ↔ 2 ≤ 1)) |
| 43 | 6, 9 | ltnlei 11382 |
. . . . . . . . . . 11
⊢ (1 < 2
↔ ¬ 2 ≤ 1) |
| 44 | | pm2.21 123 |
. . . . . . . . . . 11
⊢ (¬ 2
≤ 1 → (2 ≤ 1 → ¬ ∀𝑥 ∈ 𝐷 𝐷 = {𝑥})) |
| 45 | 43, 44 | sylbi 217 |
. . . . . . . . . 10
⊢ (1 < 2
→ (2 ≤ 1 → ¬ ∀𝑥 ∈ 𝐷 𝐷 = {𝑥})) |
| 46 | 16, 45 | ax-mp 5 |
. . . . . . . . 9
⊢ (2 ≤ 1
→ ¬ ∀𝑥
∈ 𝐷 𝐷 = {𝑥}) |
| 47 | 42, 46 | biimtrdi 253 |
. . . . . . . 8
⊢
((♯‘𝐷) =
1 → (2 ≤ (♯‘𝐷) → ¬ ∀𝑥 ∈ 𝐷 𝐷 = {𝑥})) |
| 48 | 47 | com12 32 |
. . . . . . 7
⊢ (2 ≤
(♯‘𝐷) →
((♯‘𝐷) = 1
→ ¬ ∀𝑥
∈ 𝐷 𝐷 = {𝑥})) |
| 49 | 48 | adantl 481 |
. . . . . 6
⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → ((♯‘𝐷) = 1 → ¬ ∀𝑥 ∈ 𝐷 𝐷 = {𝑥})) |
| 50 | 41, 49 | syldc 48 |
. . . . 5
⊢
(∀𝑥 ∈
𝐷 (♯‘𝐷) = 1 → ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → ¬ ∀𝑥 ∈ 𝐷 𝐷 = {𝑥})) |
| 51 | 4, 50 | syl 17 |
. . . 4
⊢
(∀𝑥 ∈
𝐷 𝐷 = {𝑥} → ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → ¬ ∀𝑥 ∈ 𝐷 𝐷 = {𝑥})) |
| 52 | | ax-1 6 |
. . . 4
⊢ (¬
∀𝑥 ∈ 𝐷 𝐷 = {𝑥} → ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → ¬ ∀𝑥 ∈ 𝐷 𝐷 = {𝑥})) |
| 53 | 51, 52 | pm2.61i 182 |
. . 3
⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → ¬ ∀𝑥 ∈ 𝐷 𝐷 = {𝑥}) |
| 54 | | eqsn 4829 |
. . . . . 6
⊢ (𝐷 ≠ ∅ → (𝐷 = {𝑥} ↔ ∀𝑦 ∈ 𝐷 𝑦 = 𝑥)) |
| 55 | 39, 54 | syl 17 |
. . . . 5
⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → (𝐷 = {𝑥} ↔ ∀𝑦 ∈ 𝐷 𝑦 = 𝑥)) |
| 56 | | equcom 2017 |
. . . . . . 7
⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) |
| 57 | 56 | a1i 11 |
. . . . . 6
⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)) |
| 58 | 57 | ralbidv 3178 |
. . . . 5
⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → (∀𝑦 ∈ 𝐷 𝑦 = 𝑥 ↔ ∀𝑦 ∈ 𝐷 𝑥 = 𝑦)) |
| 59 | 55, 58 | bitrd 279 |
. . . 4
⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → (𝐷 = {𝑥} ↔ ∀𝑦 ∈ 𝐷 𝑥 = 𝑦)) |
| 60 | 59 | ralbidv 3178 |
. . 3
⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → (∀𝑥 ∈ 𝐷 𝐷 = {𝑥} ↔ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 𝑥 = 𝑦)) |
| 61 | 53, 60 | mtbid 324 |
. 2
⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → ¬ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 𝑥 = 𝑦) |
| 62 | | df-ne 2941 |
. . . . . 6
⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦) |
| 63 | 62 | rexbii 3094 |
. . . . 5
⊢
(∃𝑦 ∈
𝐷 𝑥 ≠ 𝑦 ↔ ∃𝑦 ∈ 𝐷 ¬ 𝑥 = 𝑦) |
| 64 | | rexnal 3100 |
. . . . 5
⊢
(∃𝑦 ∈
𝐷 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐷 𝑥 = 𝑦) |
| 65 | 63, 64 | bitri 275 |
. . . 4
⊢
(∃𝑦 ∈
𝐷 𝑥 ≠ 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐷 𝑥 = 𝑦) |
| 66 | 65 | rexbii 3094 |
. . 3
⊢
(∃𝑥 ∈
𝐷 ∃𝑦 ∈ 𝐷 𝑥 ≠ 𝑦 ↔ ∃𝑥 ∈ 𝐷 ¬ ∀𝑦 ∈ 𝐷 𝑥 = 𝑦) |
| 67 | | rexnal 3100 |
. . 3
⊢
(∃𝑥 ∈
𝐷 ¬ ∀𝑦 ∈ 𝐷 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 𝑥 = 𝑦) |
| 68 | 66, 67 | bitri 275 |
. 2
⊢
(∃𝑥 ∈
𝐷 ∃𝑦 ∈ 𝐷 𝑥 ≠ 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 𝑥 = 𝑦) |
| 69 | 61, 68 | sylibr 234 |
1
⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 𝑥 ≠ 𝑦) |