Step | Hyp | Ref
| Expression |
1 | | hashbc.4 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈ ℤ) |
2 | | hashbc.3 |
. . . . 5
⊢ (𝜑 → ∀𝑗 ∈ ℤ ((#‘𝐴)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑗})) |
3 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑗 = 𝐾 → ((#‘𝐴)C𝑗) = ((#‘𝐴)C𝐾)) |
4 | | eqeq2 2621 |
. . . . . . . . 9
⊢ (𝑗 = 𝐾 → ((#‘𝑥) = 𝑗 ↔ (#‘𝑥) = 𝐾)) |
5 | 4 | rabbidv 3164 |
. . . . . . . 8
⊢ (𝑗 = 𝐾 → {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑗} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}) |
6 | 5 | fveq2d 6107 |
. . . . . . 7
⊢ (𝑗 = 𝐾 → (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑗}) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾})) |
7 | 3, 6 | eqeq12d 2625 |
. . . . . 6
⊢ (𝑗 = 𝐾 → (((#‘𝐴)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑗}) ↔ ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}))) |
8 | 7 | rspcv 3278 |
. . . . 5
⊢ (𝐾 ∈ ℤ →
(∀𝑗 ∈ ℤ
((#‘𝐴)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑗}) → ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}))) |
9 | 1, 2, 8 | sylc 63 |
. . . 4
⊢ (𝜑 → ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾})) |
10 | | ssun1 3738 |
. . . . . . . . . . . . 13
⊢ 𝐴 ⊆ (𝐴 ∪ {𝑧}) |
11 | | sspwb 4844 |
. . . . . . . . . . . . 13
⊢ (𝐴 ⊆ (𝐴 ∪ {𝑧}) ↔ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ {𝑧})) |
12 | 10, 11 | mpbi 219 |
. . . . . . . . . . . 12
⊢ 𝒫
𝐴 ⊆ 𝒫 (𝐴 ∪ {𝑧}) |
13 | 12 | sseli 3564 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧})) |
14 | 13 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧})) |
15 | | hashbc.2 |
. . . . . . . . . . 11
⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴) |
16 | | elpwi 4117 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) |
17 | 16 | ssneld 3570 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝒫 𝐴 → (¬ 𝑧 ∈ 𝐴 → ¬ 𝑧 ∈ 𝑥)) |
18 | 15, 17 | mpan9 485 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → ¬ 𝑧 ∈ 𝑥) |
19 | 14, 18 | jca 553 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥)) |
20 | | elpwi 4117 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑥 ⊆ (𝐴 ∪ {𝑧})) |
21 | | uncom 3719 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∪ {𝑧}) = ({𝑧} ∪ 𝐴) |
22 | 20, 21 | syl6sseq 3614 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑥 ⊆ ({𝑧} ∪ 𝐴)) |
23 | 22 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ⊆ ({𝑧} ∪ 𝐴)) |
24 | | simpr 476 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → ¬ 𝑧 ∈ 𝑥) |
25 | | disjsn 4192 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑥) |
26 | 24, 25 | sylibr 223 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → (𝑥 ∩ {𝑧}) = ∅) |
27 | | disjssun 3988 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∩ {𝑧}) = ∅ → (𝑥 ⊆ ({𝑧} ∪ 𝐴) ↔ 𝑥 ⊆ 𝐴)) |
28 | 26, 27 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → (𝑥 ⊆ ({𝑧} ∪ 𝐴) ↔ 𝑥 ⊆ 𝐴)) |
29 | 23, 28 | mpbid 221 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ⊆ 𝐴) |
30 | | vex 3176 |
. . . . . . . . . . . 12
⊢ 𝑥 ∈ V |
31 | 30 | elpw 4114 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
32 | 29, 31 | sylibr 223 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) → 𝑥 ∈ 𝒫 𝐴) |
33 | 32 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥)) → 𝑥 ∈ 𝒫 𝐴) |
34 | 19, 33 | impbida 873 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥))) |
35 | 34 | anbi1d 737 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ (#‘𝑥) = 𝐾) ↔ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) ∧ (#‘𝑥) = 𝐾))) |
36 | | anass 679 |
. . . . . . 7
⊢ (((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧 ∈ 𝑥) ∧ (#‘𝑥) = 𝐾) ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾))) |
37 | 35, 36 | syl6bb 275 |
. . . . . 6
⊢ (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ (#‘𝑥) = 𝐾) ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)))) |
38 | 37 | rabbidva2 3162 |
. . . . 5
⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾} = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) |
39 | 38 | fveq2d 6107 |
. . . 4
⊢ (𝜑 → (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}) = (#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)})) |
40 | 9, 39 | eqtrd 2644 |
. . 3
⊢ (𝜑 → ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)})) |
41 | | peano2zm 11297 |
. . . . . 6
⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈
ℤ) |
42 | 1, 41 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐾 − 1) ∈ ℤ) |
43 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑗 = (𝐾 − 1) → ((#‘𝐴)C𝑗) = ((#‘𝐴)C(𝐾 − 1))) |
44 | | eqeq2 2621 |
. . . . . . . . 9
⊢ (𝑗 = (𝐾 − 1) → ((#‘𝑥) = 𝑗 ↔ (#‘𝑥) = (𝐾 − 1))) |
45 | 44 | rabbidv 3164 |
. . . . . . . 8
⊢ (𝑗 = (𝐾 − 1) → {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑗} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)}) |
46 | 45 | fveq2d 6107 |
. . . . . . 7
⊢ (𝑗 = (𝐾 − 1) → (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑗}) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)})) |
47 | 43, 46 | eqeq12d 2625 |
. . . . . 6
⊢ (𝑗 = (𝐾 − 1) → (((#‘𝐴)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑗}) ↔ ((#‘𝐴)C(𝐾 − 1)) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)}))) |
48 | 47 | rspcv 3278 |
. . . . 5
⊢ ((𝐾 − 1) ∈ ℤ
→ (∀𝑗 ∈
ℤ ((#‘𝐴)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑗}) → ((#‘𝐴)C(𝐾 − 1)) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)}))) |
49 | 42, 2, 48 | sylc 63 |
. . . 4
⊢ (𝜑 → ((#‘𝐴)C(𝐾 − 1)) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)})) |
50 | | hashbc.1 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ Fin) |
51 | | pwfi 8144 |
. . . . . . . 8
⊢ (𝐴 ∈ Fin ↔ 𝒫
𝐴 ∈
Fin) |
52 | 50, 51 | sylib 207 |
. . . . . . 7
⊢ (𝜑 → 𝒫 𝐴 ∈ Fin) |
53 | | rabexg 4739 |
. . . . . . 7
⊢
(𝒫 𝐴 ∈
Fin → {𝑥 ∈
𝒫 𝐴 ∣
(#‘𝑥) = (𝐾 − 1)} ∈
V) |
54 | 52, 53 | syl 17 |
. . . . . 6
⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)} ∈ V) |
55 | | snfi 7923 |
. . . . . . . . . 10
⊢ {𝑧} ∈ Fin |
56 | | unfi 8112 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin) |
57 | 50, 55, 56 | sylancl 693 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin) |
58 | | pwfi 8144 |
. . . . . . . . 9
⊢ ((𝐴 ∪ {𝑧}) ∈ Fin ↔ 𝒫 (𝐴 ∪ {𝑧}) ∈ Fin) |
59 | 57, 58 | sylib 207 |
. . . . . . . 8
⊢ (𝜑 → 𝒫 (𝐴 ∪ {𝑧}) ∈ Fin) |
60 | | ssrab2 3650 |
. . . . . . . 8
⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧}) |
61 | | ssfi 8065 |
. . . . . . . 8
⊢
((𝒫 (𝐴 ∪
{𝑧}) ∈ Fin ∧
{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})) → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∈ Fin) |
62 | 59, 60, 61 | sylancl 693 |
. . . . . . 7
⊢ (𝜑 → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∈ Fin) |
63 | | elex 3185 |
. . . . . . 7
⊢ ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∈ Fin → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∈ V) |
64 | 62, 63 | syl 17 |
. . . . . 6
⊢ (𝜑 → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∈ V) |
65 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑥 = 𝑢 → (#‘𝑥) = (#‘𝑢)) |
66 | 65 | eqeq1d 2612 |
. . . . . . . 8
⊢ (𝑥 = 𝑢 → ((#‘𝑥) = (𝐾 − 1) ↔ (#‘𝑢) = (𝐾 − 1))) |
67 | 66 | elrab 3331 |
. . . . . . 7
⊢ (𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)} ↔ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) |
68 | | elpwi 4117 |
. . . . . . . . . . . 12
⊢ (𝑢 ∈ 𝒫 𝐴 → 𝑢 ⊆ 𝐴) |
69 | 68 | ad2antrl 760 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → 𝑢 ⊆ 𝐴) |
70 | | unss1 3744 |
. . . . . . . . . . 11
⊢ (𝑢 ⊆ 𝐴 → (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧})) |
71 | 69, 70 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧})) |
72 | | vex 3176 |
. . . . . . . . . . . 12
⊢ 𝑢 ∈ V |
73 | | snex 4835 |
. . . . . . . . . . . 12
⊢ {𝑧} ∈ V |
74 | 72, 73 | unex 6854 |
. . . . . . . . . . 11
⊢ (𝑢 ∪ {𝑧}) ∈ V |
75 | 74 | elpw 4114 |
. . . . . . . . . 10
⊢ ((𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧}) ↔ (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧})) |
76 | 71, 75 | sylibr 223 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧})) |
77 | 50 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → 𝐴 ∈ Fin) |
78 | | ssfi 8065 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ Fin ∧ 𝑢 ⊆ 𝐴) → 𝑢 ∈ Fin) |
79 | 77, 69, 78 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → 𝑢 ∈ Fin) |
80 | 55 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → {𝑧} ∈ Fin) |
81 | 15 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → ¬ 𝑧 ∈ 𝐴) |
82 | 69, 81 | ssneldd 3571 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → ¬ 𝑧 ∈ 𝑢) |
83 | | disjsn 4192 |
. . . . . . . . . . . . 13
⊢ ((𝑢 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑢) |
84 | 82, 83 | sylibr 223 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → (𝑢 ∩ {𝑧}) = ∅) |
85 | | hashun 13032 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ Fin ∧ {𝑧} ∈ Fin ∧ (𝑢 ∩ {𝑧}) = ∅) → (#‘(𝑢 ∪ {𝑧})) = ((#‘𝑢) + (#‘{𝑧}))) |
86 | 79, 80, 84, 85 | syl3anc 1318 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → (#‘(𝑢 ∪ {𝑧})) = ((#‘𝑢) + (#‘{𝑧}))) |
87 | | simprr 792 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → (#‘𝑢) = (𝐾 − 1)) |
88 | | vex 3176 |
. . . . . . . . . . . . . 14
⊢ 𝑧 ∈ V |
89 | | hashsng 13020 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ V → (#‘{𝑧}) = 1) |
90 | 88, 89 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(#‘{𝑧}) =
1 |
91 | 90 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → (#‘{𝑧}) = 1) |
92 | 87, 91 | oveq12d 6567 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → ((#‘𝑢) + (#‘{𝑧})) = ((𝐾 − 1) + 1)) |
93 | 1 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → 𝐾 ∈ ℤ) |
94 | 93 | zcnd 11359 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → 𝐾 ∈ ℂ) |
95 | | ax-1cn 9873 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℂ |
96 | | npcan 10169 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝐾 −
1) + 1) = 𝐾) |
97 | 94, 95, 96 | sylancl 693 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → ((𝐾 − 1) + 1) = 𝐾) |
98 | 86, 92, 97 | 3eqtrd 2648 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → (#‘(𝑢 ∪ {𝑧})) = 𝐾) |
99 | | ssun2 3739 |
. . . . . . . . . . 11
⊢ {𝑧} ⊆ (𝑢 ∪ {𝑧}) |
100 | 88 | snss 4259 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ (𝑢 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝑢 ∪ {𝑧})) |
101 | 99, 100 | mpbir 220 |
. . . . . . . . . 10
⊢ 𝑧 ∈ (𝑢 ∪ {𝑧}) |
102 | 98, 101 | jctil 558 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → (𝑧 ∈ (𝑢 ∪ {𝑧}) ∧ (#‘(𝑢 ∪ {𝑧})) = 𝐾)) |
103 | | eleq2 2677 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑢 ∪ {𝑧}) → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ (𝑢 ∪ {𝑧}))) |
104 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝑢 ∪ {𝑧}) → (#‘𝑥) = (#‘(𝑢 ∪ {𝑧}))) |
105 | 104 | eqeq1d 2612 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑢 ∪ {𝑧}) → ((#‘𝑥) = 𝐾 ↔ (#‘(𝑢 ∪ {𝑧})) = 𝐾)) |
106 | 103, 105 | anbi12d 743 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑢 ∪ {𝑧}) → ((𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾) ↔ (𝑧 ∈ (𝑢 ∪ {𝑧}) ∧ (#‘(𝑢 ∪ {𝑧})) = 𝐾))) |
107 | 106 | elrab 3331 |
. . . . . . . . 9
⊢ ((𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ↔ ((𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ (𝑢 ∪ {𝑧}) ∧ (#‘(𝑢 ∪ {𝑧})) = 𝐾))) |
108 | 76, 102, 107 | sylanbrc 695 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) |
109 | 108 | ex 449 |
. . . . . . 7
⊢ (𝜑 → ((𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1)) → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)})) |
110 | 67, 109 | syl5bi 231 |
. . . . . 6
⊢ (𝜑 → (𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)} → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)})) |
111 | | eleq2 2677 |
. . . . . . . . 9
⊢ (𝑥 = 𝑣 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑣)) |
112 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑣 → (#‘𝑥) = (#‘𝑣)) |
113 | 112 | eqeq1d 2612 |
. . . . . . . . 9
⊢ (𝑥 = 𝑣 → ((#‘𝑥) = 𝐾 ↔ (#‘𝑣) = 𝐾)) |
114 | 111, 113 | anbi12d 743 |
. . . . . . . 8
⊢ (𝑥 = 𝑣 → ((𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾) ↔ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) |
115 | 114 | elrab 3331 |
. . . . . . 7
⊢ (𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ↔ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) |
116 | | elpwi 4117 |
. . . . . . . . . . . . 13
⊢ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑣 ⊆ (𝐴 ∪ {𝑧})) |
117 | 116 | ad2antrl 760 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → 𝑣 ⊆ (𝐴 ∪ {𝑧})) |
118 | 117, 21 | syl6sseq 3614 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → 𝑣 ⊆ ({𝑧} ∪ 𝐴)) |
119 | | ssundif 4004 |
. . . . . . . . . . 11
⊢ (𝑣 ⊆ ({𝑧} ∪ 𝐴) ↔ (𝑣 ∖ {𝑧}) ⊆ 𝐴) |
120 | 118, 119 | sylib 207 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ⊆ 𝐴) |
121 | | vex 3176 |
. . . . . . . . . . . 12
⊢ 𝑣 ∈ V |
122 | | difexg 4735 |
. . . . . . . . . . . 12
⊢ (𝑣 ∈ V → (𝑣 ∖ {𝑧}) ∈ V) |
123 | 121, 122 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (𝑣 ∖ {𝑧}) ∈ V |
124 | 123 | elpw 4114 |
. . . . . . . . . 10
⊢ ((𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴 ↔ (𝑣 ∖ {𝑧}) ⊆ 𝐴) |
125 | 120, 124 | sylibr 223 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴) |
126 | 50 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → 𝐴 ∈ Fin) |
127 | | ssfi 8065 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ Fin ∧ (𝑣 ∖ {𝑧}) ⊆ 𝐴) → (𝑣 ∖ {𝑧}) ∈ Fin) |
128 | 126, 120,
127 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ Fin) |
129 | | hashcl 13009 |
. . . . . . . . . . . . 13
⊢ ((𝑣 ∖ {𝑧}) ∈ Fin → (#‘(𝑣 ∖ {𝑧})) ∈
ℕ0) |
130 | 128, 129 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (#‘(𝑣 ∖ {𝑧})) ∈
ℕ0) |
131 | 130 | nn0cnd 11230 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (#‘(𝑣 ∖ {𝑧})) ∈ ℂ) |
132 | | pncan 10166 |
. . . . . . . . . . 11
⊢
(((#‘(𝑣
∖ {𝑧})) ∈
ℂ ∧ 1 ∈ ℂ) → (((#‘(𝑣 ∖ {𝑧})) + 1) − 1) = (#‘(𝑣 ∖ {𝑧}))) |
133 | 131, 95, 132 | sylancl 693 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (((#‘(𝑣 ∖ {𝑧})) + 1) − 1) = (#‘(𝑣 ∖ {𝑧}))) |
134 | | undif1 3995 |
. . . . . . . . . . . . . 14
⊢ ((𝑣 ∖ {𝑧}) ∪ {𝑧}) = (𝑣 ∪ {𝑧}) |
135 | | simprrl 800 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → 𝑧 ∈ 𝑣) |
136 | 135 | snssd 4281 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → {𝑧} ⊆ 𝑣) |
137 | | ssequn2 3748 |
. . . . . . . . . . . . . . 15
⊢ ({𝑧} ⊆ 𝑣 ↔ (𝑣 ∪ {𝑧}) = 𝑣) |
138 | 136, 137 | sylib 207 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (𝑣 ∪ {𝑧}) = 𝑣) |
139 | 134, 138 | syl5eq 2656 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) ∪ {𝑧}) = 𝑣) |
140 | 139 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (#‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = (#‘𝑣)) |
141 | 55 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → {𝑧} ∈ Fin) |
142 | | incom 3767 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ({𝑧} ∩ (𝑣 ∖ {𝑧})) |
143 | | disjdif 3992 |
. . . . . . . . . . . . . . . 16
⊢ ({𝑧} ∩ (𝑣 ∖ {𝑧})) = ∅ |
144 | 142, 143 | eqtri 2632 |
. . . . . . . . . . . . . . 15
⊢ ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅ |
145 | 144 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅) |
146 | | hashun 13032 |
. . . . . . . . . . . . . 14
⊢ (((𝑣 ∖ {𝑧}) ∈ Fin ∧ {𝑧} ∈ Fin ∧ ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (#‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((#‘(𝑣 ∖ {𝑧})) + (#‘{𝑧}))) |
147 | 128, 141,
145, 146 | syl3anc 1318 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (#‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((#‘(𝑣 ∖ {𝑧})) + (#‘{𝑧}))) |
148 | 90 | oveq2i 6560 |
. . . . . . . . . . . . 13
⊢
((#‘(𝑣 ∖
{𝑧})) + (#‘{𝑧})) = ((#‘(𝑣 ∖ {𝑧})) + 1) |
149 | 147, 148 | syl6eq 2660 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (#‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((#‘(𝑣 ∖ {𝑧})) + 1)) |
150 | | simprrr 801 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (#‘𝑣) = 𝐾) |
151 | 140, 149,
150 | 3eqtr3d 2652 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → ((#‘(𝑣 ∖ {𝑧})) + 1) = 𝐾) |
152 | 151 | oveq1d 6564 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (((#‘(𝑣 ∖ {𝑧})) + 1) − 1) = (𝐾 − 1)) |
153 | 133, 152 | eqtr3d 2646 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (#‘(𝑣 ∖ {𝑧})) = (𝐾 − 1)) |
154 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑣 ∖ {𝑧}) → (#‘𝑥) = (#‘(𝑣 ∖ {𝑧}))) |
155 | 154 | eqeq1d 2612 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑣 ∖ {𝑧}) → ((#‘𝑥) = (𝐾 − 1) ↔ (#‘(𝑣 ∖ {𝑧})) = (𝐾 − 1))) |
156 | 155 | elrab 3331 |
. . . . . . . . 9
⊢ ((𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)} ↔ ((𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴 ∧ (#‘(𝑣 ∖ {𝑧})) = (𝐾 − 1))) |
157 | 125, 153,
156 | sylanbrc 695 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)}) |
158 | 157 | ex 449 |
. . . . . . 7
⊢ (𝜑 → ((𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾)) → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)})) |
159 | 115, 158 | syl5bi 231 |
. . . . . 6
⊢ (𝜑 → (𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)})) |
160 | 67, 115 | anbi12i 729 |
. . . . . . 7
⊢ ((𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)} ∧ 𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) ↔ ((𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾)))) |
161 | | simp3rl 1127 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → 𝑧 ∈ 𝑣) |
162 | 161 | snssd 4281 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → {𝑧} ⊆ 𝑣) |
163 | | incom 3767 |
. . . . . . . . . . . 12
⊢ ({𝑧} ∩ 𝑢) = (𝑢 ∩ {𝑧}) |
164 | 84 | 3adant3 1074 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (𝑢 ∩ {𝑧}) = ∅) |
165 | 163, 164 | syl5eq 2656 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → ({𝑧} ∩ 𝑢) = ∅) |
166 | | uneqdifeq 4009 |
. . . . . . . . . . 11
⊢ (({𝑧} ⊆ 𝑣 ∧ ({𝑧} ∩ 𝑢) = ∅) → (({𝑧} ∪ 𝑢) = 𝑣 ↔ (𝑣 ∖ {𝑧}) = 𝑢)) |
167 | 162, 165,
166 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (({𝑧} ∪ 𝑢) = 𝑣 ↔ (𝑣 ∖ {𝑧}) = 𝑢)) |
168 | 167 | bicomd 212 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) = 𝑢 ↔ ({𝑧} ∪ 𝑢) = 𝑣)) |
169 | | eqcom 2617 |
. . . . . . . . 9
⊢ (𝑢 = (𝑣 ∖ {𝑧}) ↔ (𝑣 ∖ {𝑧}) = 𝑢) |
170 | | eqcom 2617 |
. . . . . . . . . 10
⊢ (𝑣 = (𝑢 ∪ {𝑧}) ↔ (𝑢 ∪ {𝑧}) = 𝑣) |
171 | | uncom 3719 |
. . . . . . . . . . 11
⊢ (𝑢 ∪ {𝑧}) = ({𝑧} ∪ 𝑢) |
172 | 171 | eqeq1i 2615 |
. . . . . . . . . 10
⊢ ((𝑢 ∪ {𝑧}) = 𝑣 ↔ ({𝑧} ∪ 𝑢) = 𝑣) |
173 | 170, 172 | bitri 263 |
. . . . . . . . 9
⊢ (𝑣 = (𝑢 ∪ {𝑧}) ↔ ({𝑧} ∪ 𝑢) = 𝑣) |
174 | 168, 169,
173 | 3bitr4g 302 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧}))) |
175 | 174 | 3expib 1260 |
. . . . . . 7
⊢ (𝜑 → (((𝑢 ∈ 𝒫 𝐴 ∧ (#‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧 ∈ 𝑣 ∧ (#‘𝑣) = 𝐾))) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧})))) |
176 | 160, 175 | syl5bi 231 |
. . . . . 6
⊢ (𝜑 → ((𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)} ∧ 𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧})))) |
177 | 54, 64, 110, 159, 176 | en3d 7878 |
. . . . 5
⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)} ≈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) |
178 | | ssrab2 3650 |
. . . . . . 7
⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)} ⊆ 𝒫 𝐴 |
179 | | ssfi 8065 |
. . . . . . 7
⊢
((𝒫 𝐴 ∈
Fin ∧ {𝑥 ∈
𝒫 𝐴 ∣
(#‘𝑥) = (𝐾 − 1)} ⊆ 𝒫
𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)} ∈ Fin) |
180 | 52, 178, 179 | sylancl 693 |
. . . . . 6
⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)} ∈ Fin) |
181 | | hashen 12997 |
. . . . . 6
⊢ (({𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)} ∈ Fin ∧ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∈ Fin) → ((#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)}) = (#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) ↔ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)} ≈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)})) |
182 | 180, 62, 181 | syl2anc 691 |
. . . . 5
⊢ (𝜑 → ((#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)}) = (#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) ↔ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)} ≈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)})) |
183 | 177, 182 | mpbird 246 |
. . . 4
⊢ (𝜑 → (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = (𝐾 − 1)}) = (#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)})) |
184 | 49, 183 | eqtrd 2644 |
. . 3
⊢ (𝜑 → ((#‘𝐴)C(𝐾 − 1)) = (#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)})) |
185 | 40, 184 | oveq12d 6567 |
. 2
⊢ (𝜑 → (((#‘𝐴)C𝐾) + ((#‘𝐴)C(𝐾 − 1))) = ((#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) + (#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}))) |
186 | 55 | a1i 11 |
. . . . . 6
⊢ (𝜑 → {𝑧} ∈ Fin) |
187 | | disjsn 4192 |
. . . . . . 7
⊢ ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝐴) |
188 | 15, 187 | sylibr 223 |
. . . . . 6
⊢ (𝜑 → (𝐴 ∩ {𝑧}) = ∅) |
189 | | hashun 13032 |
. . . . . 6
⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin ∧ (𝐴 ∩ {𝑧}) = ∅) → (#‘(𝐴 ∪ {𝑧})) = ((#‘𝐴) + (#‘{𝑧}))) |
190 | 50, 186, 188, 189 | syl3anc 1318 |
. . . . 5
⊢ (𝜑 → (#‘(𝐴 ∪ {𝑧})) = ((#‘𝐴) + (#‘{𝑧}))) |
191 | 90 | oveq2i 6560 |
. . . . 5
⊢
((#‘𝐴) +
(#‘{𝑧})) =
((#‘𝐴) +
1) |
192 | 190, 191 | syl6eq 2660 |
. . . 4
⊢ (𝜑 → (#‘(𝐴 ∪ {𝑧})) = ((#‘𝐴) + 1)) |
193 | 192 | oveq1d 6564 |
. . 3
⊢ (𝜑 → ((#‘(𝐴 ∪ {𝑧}))C𝐾) = (((#‘𝐴) + 1)C𝐾)) |
194 | | hashcl 13009 |
. . . . 5
⊢ (𝐴 ∈ Fin →
(#‘𝐴) ∈
ℕ0) |
195 | 50, 194 | syl 17 |
. . . 4
⊢ (𝜑 → (#‘𝐴) ∈
ℕ0) |
196 | | bcpasc 12970 |
. . . 4
⊢
(((#‘𝐴) ∈
ℕ0 ∧ 𝐾
∈ ℤ) → (((#‘𝐴)C𝐾) + ((#‘𝐴)C(𝐾 − 1))) = (((#‘𝐴) + 1)C𝐾)) |
197 | 195, 1, 196 | syl2anc 691 |
. . 3
⊢ (𝜑 → (((#‘𝐴)C𝐾) + ((#‘𝐴)C(𝐾 − 1))) = (((#‘𝐴) + 1)C𝐾)) |
198 | 193, 197 | eqtr4d 2647 |
. 2
⊢ (𝜑 → ((#‘(𝐴 ∪ {𝑧}))C𝐾) = (((#‘𝐴)C𝐾) + ((#‘𝐴)C(𝐾 − 1)))) |
199 | | pm2.1 432 |
. . . . . . . . 9
⊢ (¬
𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥) |
200 | 199 | biantrur 526 |
. . . . . . . 8
⊢
((#‘𝑥) = 𝐾 ↔ ((¬ 𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥) ∧ (#‘𝑥) = 𝐾)) |
201 | | andir 908 |
. . . . . . . 8
⊢ (((¬
𝑧 ∈ 𝑥 ∨ 𝑧 ∈ 𝑥) ∧ (#‘𝑥) = 𝐾) ↔ ((¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾))) |
202 | 200, 201 | bitri 263 |
. . . . . . 7
⊢
((#‘𝑥) = 𝐾 ↔ ((¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾))) |
203 | 202 | a1i 11 |
. . . . . 6
⊢ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) → ((#‘𝑥) = 𝐾 ↔ ((¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)))) |
204 | 203 | rabbiia 3161 |
. . . . 5
⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (#‘𝑥) = 𝐾} = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾))} |
205 | | unrab 3857 |
. . . . 5
⊢ ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾) ∨ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾))} |
206 | 204, 205 | eqtr4i 2635 |
. . . 4
⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (#‘𝑥) = 𝐾} = ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) |
207 | 206 | fveq2i 6106 |
. . 3
⊢
(#‘{𝑥 ∈
𝒫 (𝐴 ∪ {𝑧}) ∣ (#‘𝑥) = 𝐾}) = (#‘({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)})) |
208 | | ssrab2 3650 |
. . . . 5
⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧}) |
209 | | ssfi 8065 |
. . . . 5
⊢
((𝒫 (𝐴 ∪
{𝑧}) ∈ Fin ∧
{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})) → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∈ Fin) |
210 | 59, 208, 209 | sylancl 693 |
. . . 4
⊢ (𝜑 → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∈ Fin) |
211 | | inrab 3858 |
. . . . . 6
⊢ ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾))} |
212 | | simprl 790 |
. . . . . . . . 9
⊢ (((¬
𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)) → 𝑧 ∈ 𝑥) |
213 | | simpll 786 |
. . . . . . . . 9
⊢ (((¬
𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)) → ¬ 𝑧 ∈ 𝑥) |
214 | 212, 213 | pm2.65i 184 |
. . . . . . . 8
⊢ ¬
((¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)) |
215 | 214 | rgenw 2908 |
. . . . . . 7
⊢
∀𝑥 ∈
𝒫 (𝐴 ∪ {𝑧}) ¬ ((¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)) |
216 | | rabeq0 3911 |
. . . . . . 7
⊢ ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾))} = ∅ ↔ ∀𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ¬ ((¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾))) |
217 | 215, 216 | mpbir 220 |
. . . . . 6
⊢ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾) ∧ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾))} = ∅ |
218 | 211, 217 | eqtri 2632 |
. . . . 5
⊢ ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) = ∅ |
219 | 218 | a1i 11 |
. . . 4
⊢ (𝜑 → ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) = ∅) |
220 | | hashun 13032 |
. . . 4
⊢ (({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∈ Fin ∧ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∈ Fin ∧ ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) = ∅) → (#‘({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)})) = ((#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) + (#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}))) |
221 | 210, 62, 219, 220 | syl3anc 1318 |
. . 3
⊢ (𝜑 → (#‘({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)})) = ((#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) + (#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}))) |
222 | 207, 221 | syl5eq 2656 |
. 2
⊢ (𝜑 → (#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (#‘𝑥) = 𝐾}) = ((#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}) + (#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧 ∈ 𝑥 ∧ (#‘𝑥) = 𝐾)}))) |
223 | 185, 198,
222 | 3eqtr4d 2654 |
1
⊢ (𝜑 → ((#‘(𝐴 ∪ {𝑧}))C𝐾) = (#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (#‘𝑥) = 𝐾})) |