Step | Hyp | Ref
| Expression |
1 | | simp2 987 |
. 2
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) |
2 | | simp1 986 |
. 2
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐴 ∈ Fin) |
3 | | simp3 988 |
. 2
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴) |
4 | | sseq1 3161 |
. . . . . 6
⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴)) |
5 | 4 | anbi2d 460 |
. . . . 5
⊢ (𝑤 = ∅ → ((𝐴 ∈ Fin ∧ 𝑤 ⊆ 𝐴) ↔ (𝐴 ∈ Fin ∧ ∅ ⊆ 𝐴))) |
6 | | difeq2 3230 |
. . . . . 6
⊢ (𝑤 = ∅ → (𝐴 ∖ 𝑤) = (𝐴 ∖ ∅)) |
7 | 6 | eleq1d 2233 |
. . . . 5
⊢ (𝑤 = ∅ → ((𝐴 ∖ 𝑤) ∈ Fin ↔ (𝐴 ∖ ∅) ∈
Fin)) |
8 | 5, 7 | imbi12d 233 |
. . . 4
⊢ (𝑤 = ∅ → (((𝐴 ∈ Fin ∧ 𝑤 ⊆ 𝐴) → (𝐴 ∖ 𝑤) ∈ Fin) ↔ ((𝐴 ∈ Fin ∧ ∅ ⊆ 𝐴) → (𝐴 ∖ ∅) ∈
Fin))) |
9 | | sseq1 3161 |
. . . . . 6
⊢ (𝑤 = 𝑦 → (𝑤 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴)) |
10 | 9 | anbi2d 460 |
. . . . 5
⊢ (𝑤 = 𝑦 → ((𝐴 ∈ Fin ∧ 𝑤 ⊆ 𝐴) ↔ (𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴))) |
11 | | difeq2 3230 |
. . . . . 6
⊢ (𝑤 = 𝑦 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝑦)) |
12 | 11 | eleq1d 2233 |
. . . . 5
⊢ (𝑤 = 𝑦 → ((𝐴 ∖ 𝑤) ∈ Fin ↔ (𝐴 ∖ 𝑦) ∈ Fin)) |
13 | 10, 12 | imbi12d 233 |
. . . 4
⊢ (𝑤 = 𝑦 → (((𝐴 ∈ Fin ∧ 𝑤 ⊆ 𝐴) → (𝐴 ∖ 𝑤) ∈ Fin) ↔ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin))) |
14 | | sseq1 3161 |
. . . . . 6
⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ⊆ 𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) |
15 | 14 | anbi2d 460 |
. . . . 5
⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐴 ∈ Fin ∧ 𝑤 ⊆ 𝐴) ↔ (𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴))) |
16 | | difeq2 3230 |
. . . . . 6
⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐴 ∖ 𝑤) = (𝐴 ∖ (𝑦 ∪ {𝑧}))) |
17 | 16 | eleq1d 2233 |
. . . . 5
⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐴 ∖ 𝑤) ∈ Fin ↔ (𝐴 ∖ (𝑦 ∪ {𝑧})) ∈ Fin)) |
18 | 15, 17 | imbi12d 233 |
. . . 4
⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (((𝐴 ∈ Fin ∧ 𝑤 ⊆ 𝐴) → (𝐴 ∖ 𝑤) ∈ Fin) ↔ ((𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (𝐴 ∖ (𝑦 ∪ {𝑧})) ∈ Fin))) |
19 | | sseq1 3161 |
. . . . . 6
⊢ (𝑤 = 𝐵 → (𝑤 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴)) |
20 | 19 | anbi2d 460 |
. . . . 5
⊢ (𝑤 = 𝐵 → ((𝐴 ∈ Fin ∧ 𝑤 ⊆ 𝐴) ↔ (𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴))) |
21 | | difeq2 3230 |
. . . . . 6
⊢ (𝑤 = 𝐵 → (𝐴 ∖ 𝑤) = (𝐴 ∖ 𝐵)) |
22 | 21 | eleq1d 2233 |
. . . . 5
⊢ (𝑤 = 𝐵 → ((𝐴 ∖ 𝑤) ∈ Fin ↔ (𝐴 ∖ 𝐵) ∈ Fin)) |
23 | 20, 22 | imbi12d 233 |
. . . 4
⊢ (𝑤 = 𝐵 → (((𝐴 ∈ Fin ∧ 𝑤 ⊆ 𝐴) → (𝐴 ∖ 𝑤) ∈ Fin) ↔ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∖ 𝐵) ∈ Fin))) |
24 | | dif0 3475 |
. . . . . . 7
⊢ (𝐴 ∖ ∅) = 𝐴 |
25 | 24 | eleq1i 2230 |
. . . . . 6
⊢ ((𝐴 ∖ ∅) ∈ Fin
↔ 𝐴 ∈
Fin) |
26 | 25 | biimpri 132 |
. . . . 5
⊢ (𝐴 ∈ Fin → (𝐴 ∖ ∅) ∈
Fin) |
27 | 26 | adantr 274 |
. . . 4
⊢ ((𝐴 ∈ Fin ∧ ∅
⊆ 𝐴) → (𝐴 ∖ ∅) ∈
Fin) |
28 | | difun1 3378 |
. . . . . 6
⊢ (𝐴 ∖ (𝑦 ∪ {𝑧})) = ((𝐴 ∖ 𝑦) ∖ {𝑧}) |
29 | | simprl 521 |
. . . . . . . 8
⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin)) ∧ (𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝐴 ∈ Fin) |
30 | | simprr 522 |
. . . . . . . . 9
⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin)) ∧ (𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴) |
31 | 30 | unssad 3295 |
. . . . . . . 8
⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin)) ∧ (𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑦 ⊆ 𝐴) |
32 | | simplr 520 |
. . . . . . . 8
⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin)) ∧ (𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin)) |
33 | 29, 31, 32 | mp2and 430 |
. . . . . . 7
⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin)) ∧ (𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐴 ∖ 𝑦) ∈ Fin) |
34 | | vsnid 3603 |
. . . . . . . . . 10
⊢ 𝑧 ∈ {𝑧} |
35 | | simprr 522 |
. . . . . . . . . . . 12
⊢ (((𝑦 ∈ Fin ∧ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin)) ∧ (𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴) |
36 | 35 | unssbd 3296 |
. . . . . . . . . . 11
⊢ (((𝑦 ∈ Fin ∧ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin)) ∧ (𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → {𝑧} ⊆ 𝐴) |
37 | 36 | sseld 3137 |
. . . . . . . . . 10
⊢ (((𝑦 ∈ Fin ∧ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin)) ∧ (𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑧 ∈ {𝑧} → 𝑧 ∈ 𝐴)) |
38 | 34, 37 | mpi 15 |
. . . . . . . . 9
⊢ (((𝑦 ∈ Fin ∧ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin)) ∧ (𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ 𝐴) |
39 | 38 | adantllr 473 |
. . . . . . . 8
⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin)) ∧ (𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ 𝐴) |
40 | | simpllr 524 |
. . . . . . . 8
⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin)) ∧ (𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ¬ 𝑧 ∈ 𝑦) |
41 | 39, 40 | eldifd 3122 |
. . . . . . 7
⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin)) ∧ (𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ (𝐴 ∖ 𝑦)) |
42 | | diffisn 6851 |
. . . . . . 7
⊢ (((𝐴 ∖ 𝑦) ∈ Fin ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ∈ Fin) |
43 | 33, 41, 42 | syl2anc 409 |
. . . . . 6
⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin)) ∧ (𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ((𝐴 ∖ 𝑦) ∖ {𝑧}) ∈ Fin) |
44 | 28, 43 | eqeltrid 2251 |
. . . . 5
⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin)) ∧ (𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐴 ∖ (𝑦 ∪ {𝑧})) ∈ Fin) |
45 | 44 | exp31 362 |
. . . 4
⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → (𝐴 ∖ 𝑦) ∈ Fin) → ((𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (𝐴 ∖ (𝑦 ∪ {𝑧})) ∈ Fin))) |
46 | 8, 13, 18, 23, 27, 45 | findcard2s 6848 |
. . 3
⊢ (𝐵 ∈ Fin → ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∖ 𝐵) ∈ Fin)) |
47 | 46 | imp 123 |
. 2
⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴)) → (𝐴 ∖ 𝐵) ∈ Fin) |
48 | 1, 2, 3, 47 | syl12anc 1225 |
1
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∖ 𝐵) ∈ Fin) |