Step | Hyp | Ref
| Expression |
1 | | isfi 7979 |
. . 3
⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
2 | | relen 7960 |
. . . . . . . . 9
⊢ Rel
≈ |
3 | 2 | brrelexi 5158 |
. . . . . . . 8
⊢ (𝐴 ≈ 𝑥 → 𝐴 ∈ V) |
4 | | pssss 3702 |
. . . . . . . 8
⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴) |
5 | | ssdomg 8001 |
. . . . . . . . 9
⊢ (𝐴 ∈ V → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) |
6 | 5 | imp 445 |
. . . . . . . 8
⊢ ((𝐴 ∈ V ∧ 𝐵 ⊆ 𝐴) → 𝐵 ≼ 𝐴) |
7 | 3, 4, 6 | syl2an 494 |
. . . . . . 7
⊢ ((𝐴 ≈ 𝑥 ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≼ 𝐴) |
8 | 7 | adantll 750 |
. . . . . 6
⊢ (((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≼ 𝐴) |
9 | | bren 7964 |
. . . . . . . . 9
⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥) |
10 | | imass2 5501 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐵 ⊆ 𝐴 → (𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴)) |
11 | 4, 10 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝐵 ⊊ 𝐴 → (𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴)) |
12 | 11 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴)) |
13 | | pssnel 4039 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐵 ⊊ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)) |
14 | | eldif 3584 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)) |
15 | | f1ofn 6138 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓 Fn 𝐴) |
16 | | difss 3737 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 |
17 | | fnfvima 6496 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑓 Fn 𝐴 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ 𝑦 ∈ (𝐴 ∖ 𝐵)) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵))) |
18 | 17 | 3expia 1267 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑓 Fn 𝐴 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴) → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵)))) |
19 | 15, 16, 18 | sylancl 694 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵)))) |
20 | | dff1o3 6143 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑓:𝐴–1-1-onto→𝑥 ↔ (𝑓:𝐴–onto→𝑥 ∧ Fun ◡𝑓)) |
21 | 20 | simprbi 480 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑓:𝐴–1-1-onto→𝑥 → Fun ◡𝑓) |
22 | | imadif 5973 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (Fun
◡𝑓 → (𝑓 “ (𝐴 ∖ 𝐵)) = ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵))) |
23 | 21, 22 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ (𝐴 ∖ 𝐵)) = ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵))) |
24 | 23 | eleq2d 2687 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑓:𝐴–1-1-onto→𝑥 → ((𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵)) ↔ (𝑓‘𝑦) ∈ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)))) |
25 | 19, 24 | sylibd 229 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)))) |
26 | | n0i 3920 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑓‘𝑦) ∈ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅) |
27 | 25, 26 | syl6 35 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑦 ∈ (𝐴 ∖ 𝐵) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)) |
28 | 14, 27 | syl5bir 233 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓:𝐴–1-1-onto→𝑥 → ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)) |
29 | 28 | exlimdv 1861 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:𝐴–1-1-onto→𝑥 → (∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)) |
30 | 29 | imp 445 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅) |
31 | 13, 30 | sylan2 491 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅) |
32 | | ssdif0 3942 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵) ↔ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅) |
33 | 31, 32 | sylnibr 319 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ¬ (𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵)) |
34 | | dfpss3 3693 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ ((𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴) ∧ ¬ (𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵))) |
35 | 12, 33, 34 | sylanbrc 698 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴)) |
36 | | imadmrn 5476 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 “ dom 𝑓) = ran 𝑓 |
37 | | f1odm 6141 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:𝐴–1-1-onto→𝑥 → dom 𝑓 = 𝐴) |
38 | 37 | imaeq2d 5466 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ dom 𝑓) = (𝑓 “ 𝐴)) |
39 | | f1ofo 6144 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–onto→𝑥) |
40 | | forn 6118 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:𝐴–onto→𝑥 → ran 𝑓 = 𝑥) |
41 | 39, 40 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:𝐴–1-1-onto→𝑥 → ran 𝑓 = 𝑥) |
42 | 36, 38, 41 | 3eqtr3a 2680 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ 𝐴) = 𝑥) |
43 | 42 | psseq2d 3700 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:𝐴–1-1-onto→𝑥 → ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐵) ⊊ 𝑥)) |
44 | 43 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐵) ⊊ 𝑥)) |
45 | 35, 44 | mpbid 222 |
. . . . . . . . . . . . 13
⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊊ 𝑥) |
46 | | php 8144 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ω ∧ (𝑓 “ 𝐵) ⊊ 𝑥) → ¬ 𝑥 ≈ (𝑓 “ 𝐵)) |
47 | 45, 46 | sylan2 491 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ω ∧ (𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴)) → ¬ 𝑥 ≈ (𝑓 “ 𝐵)) |
48 | | f1of1 6136 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–1-1→𝑥) |
49 | | f1ores 6151 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:𝐴–1-1→𝑥 ∧ 𝐵 ⊆ 𝐴) → (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵)) |
50 | 48, 4, 49 | syl2an 494 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵)) |
51 | | vex 3203 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑓 ∈ V |
52 | 51 | resex 5443 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 ↾ 𝐵) ∈ V |
53 | | f1oeq1 6127 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = (𝑓 ↾ 𝐵) → (𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵) ↔ (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵))) |
54 | 52, 53 | spcev 3300 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵) → ∃𝑦 𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵)) |
55 | | bren 7964 |
. . . . . . . . . . . . . . . 16
⊢ (𝐵 ≈ (𝑓 “ 𝐵) ↔ ∃𝑦 𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵)) |
56 | 54, 55 | sylibr 224 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵) → 𝐵 ≈ (𝑓 “ 𝐵)) |
57 | 50, 56 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≈ (𝑓 “ 𝐵)) |
58 | | entr 8008 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ≈ 𝐵 ∧ 𝐵 ≈ (𝑓 “ 𝐵)) → 𝑥 ≈ (𝑓 “ 𝐵)) |
59 | 58 | expcom 451 |
. . . . . . . . . . . . . 14
⊢ (𝐵 ≈ (𝑓 “ 𝐵) → (𝑥 ≈ 𝐵 → 𝑥 ≈ (𝑓 “ 𝐵))) |
60 | 57, 59 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑥 ≈ 𝐵 → 𝑥 ≈ (𝑓 “ 𝐵))) |
61 | 60 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ω ∧ (𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴)) → (𝑥 ≈ 𝐵 → 𝑥 ≈ (𝑓 “ 𝐵))) |
62 | 47, 61 | mtod 189 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ω ∧ (𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴)) → ¬ 𝑥 ≈ 𝐵) |
63 | 62 | exp32 631 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ω → (𝑓:𝐴–1-1-onto→𝑥 → (𝐵 ⊊ 𝐴 → ¬ 𝑥 ≈ 𝐵))) |
64 | 63 | exlimdv 1861 |
. . . . . . . . 9
⊢ (𝑥 ∈ ω →
(∃𝑓 𝑓:𝐴–1-1-onto→𝑥 → (𝐵 ⊊ 𝐴 → ¬ 𝑥 ≈ 𝐵))) |
65 | 9, 64 | syl5bi 232 |
. . . . . . . 8
⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐵 ⊊ 𝐴 → ¬ 𝑥 ≈ 𝐵))) |
66 | 65 | imp31 448 |
. . . . . . 7
⊢ (((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) ∧ 𝐵 ⊊ 𝐴) → ¬ 𝑥 ≈ 𝐵) |
67 | | entr 8008 |
. . . . . . . . . 10
⊢ ((𝐵 ≈ 𝐴 ∧ 𝐴 ≈ 𝑥) → 𝐵 ≈ 𝑥) |
68 | 67 | ex 450 |
. . . . . . . . 9
⊢ (𝐵 ≈ 𝐴 → (𝐴 ≈ 𝑥 → 𝐵 ≈ 𝑥)) |
69 | | ensym 8005 |
. . . . . . . . 9
⊢ (𝐵 ≈ 𝑥 → 𝑥 ≈ 𝐵) |
70 | 68, 69 | syl6com 37 |
. . . . . . . 8
⊢ (𝐴 ≈ 𝑥 → (𝐵 ≈ 𝐴 → 𝑥 ≈ 𝐵)) |
71 | 70 | ad2antlr 763 |
. . . . . . 7
⊢ (((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) ∧ 𝐵 ⊊ 𝐴) → (𝐵 ≈ 𝐴 → 𝑥 ≈ 𝐵)) |
72 | 66, 71 | mtod 189 |
. . . . . 6
⊢ (((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐵 ≈ 𝐴) |
73 | | brsdom 7978 |
. . . . . 6
⊢ (𝐵 ≺ 𝐴 ↔ (𝐵 ≼ 𝐴 ∧ ¬ 𝐵 ≈ 𝐴)) |
74 | 8, 72, 73 | sylanbrc 698 |
. . . . 5
⊢ (((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴) |
75 | 74 | exp31 630 |
. . . 4
⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴))) |
76 | 75 | rexlimiv 3027 |
. . 3
⊢
(∃𝑥 ∈
ω 𝐴 ≈ 𝑥 → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴)) |
77 | 1, 76 | sylbi 207 |
. 2
⊢ (𝐴 ∈ Fin → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴)) |
78 | 77 | imp 445 |
1
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴) |