Step | Hyp | Ref
| Expression |
1 | | sdomdom 7869 |
. . . . 5
⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) |
2 | | relsdom 7848 |
. . . . . . 7
⊢ Rel
≺ |
3 | 2 | brrelex2i 5083 |
. . . . . 6
⊢ (𝐴 ≺ 𝐵 → 𝐵 ∈ V) |
4 | | brdomg 7851 |
. . . . . 6
⊢ (𝐵 ∈ V → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
5 | 3, 4 | syl 17 |
. . . . 5
⊢ (𝐴 ≺ 𝐵 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
6 | 1, 5 | mpbid 221 |
. . . 4
⊢ (𝐴 ≺ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) |
7 | 6 | adantr 480 |
. . 3
⊢ ((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) → ∃𝑓 𝑓:𝐴–1-1→𝐵) |
8 | | f1f 6014 |
. . . . . . . 8
⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) |
9 | | frn 5966 |
. . . . . . . 8
⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵) |
10 | 8, 9 | syl 17 |
. . . . . . 7
⊢ (𝑓:𝐴–1-1→𝐵 → ran 𝑓 ⊆ 𝐵) |
11 | 10 | adantl 481 |
. . . . . 6
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ⊆ 𝐵) |
12 | | sdomnen 7870 |
. . . . . . . 8
⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) |
13 | 12 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ¬ 𝐴 ≈ 𝐵) |
14 | | vex 3176 |
. . . . . . . . . . 11
⊢ 𝑓 ∈ V |
15 | | dff1o5 6059 |
. . . . . . . . . . . 12
⊢ (𝑓:𝐴–1-1-onto→𝐵 ↔ (𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵)) |
16 | 15 | biimpri 217 |
. . . . . . . . . . 11
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵) → 𝑓:𝐴–1-1-onto→𝐵) |
17 | | f1oen3g 7857 |
. . . . . . . . . . 11
⊢ ((𝑓 ∈ V ∧ 𝑓:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) |
18 | 14, 16, 17 | sylancr 694 |
. . . . . . . . . 10
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵) → 𝐴 ≈ 𝐵) |
19 | 18 | ex 449 |
. . . . . . . . 9
⊢ (𝑓:𝐴–1-1→𝐵 → (ran 𝑓 = 𝐵 → 𝐴 ≈ 𝐵)) |
20 | 19 | necon3bd 2796 |
. . . . . . . 8
⊢ (𝑓:𝐴–1-1→𝐵 → (¬ 𝐴 ≈ 𝐵 → ran 𝑓 ≠ 𝐵)) |
21 | 20 | adantl 481 |
. . . . . . 7
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (¬ 𝐴 ≈ 𝐵 → ran 𝑓 ≠ 𝐵)) |
22 | 13, 21 | mpd 15 |
. . . . . 6
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ≠ 𝐵) |
23 | | pssdifn0 3898 |
. . . . . 6
⊢ ((ran
𝑓 ⊆ 𝐵 ∧ ran 𝑓 ≠ 𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅) |
24 | 11, 22, 23 | syl2anc 691 |
. . . . 5
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅) |
25 | | n0 3890 |
. . . . 5
⊢ ((𝐵 ∖ ran 𝑓) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓)) |
26 | 24, 25 | sylib 207 |
. . . 4
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓)) |
27 | 2 | brrelexi 5082 |
. . . . . . . . 9
⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V) |
28 | 27 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ∈ V) |
29 | 3 | ad2antrr 758 |
. . . . . . . . 9
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐵 ∈ V) |
30 | | difexg 4735 |
. . . . . . . . 9
⊢ (𝐵 ∈ V → (𝐵 ∖ {𝑥}) ∈ V) |
31 | 29, 30 | syl 17 |
. . . . . . . 8
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ∈ V) |
32 | | eldifn 3695 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → ¬ 𝑥 ∈ ran 𝑓) |
33 | | disjsn 4192 |
. . . . . . . . . . . . 13
⊢ ((ran
𝑓 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝑓) |
34 | 32, 33 | sylibr 223 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → (ran 𝑓 ∩ {𝑥}) = ∅) |
35 | 34 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → (ran 𝑓 ∩ {𝑥}) = ∅) |
36 | 10 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ran 𝑓 ⊆ 𝐵) |
37 | | reldisj 3972 |
. . . . . . . . . . . 12
⊢ (ran
𝑓 ⊆ 𝐵 → ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ran 𝑓 ⊆ (𝐵 ∖ {𝑥}))) |
38 | 36, 37 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ran 𝑓 ⊆ (𝐵 ∖ {𝑥}))) |
39 | 35, 38 | mpbid 221 |
. . . . . . . . . 10
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ran 𝑓 ⊆ (𝐵 ∖ {𝑥})) |
40 | | f1ssr 6020 |
. . . . . . . . . 10
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})) → 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥})) |
41 | 39, 40 | syldan 486 |
. . . . . . . . 9
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥})) |
42 | 41 | adantl 481 |
. . . . . . . 8
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥})) |
43 | | f1dom2g 7859 |
. . . . . . . 8
⊢ ((𝐴 ∈ V ∧ (𝐵 ∖ {𝑥}) ∈ V ∧ 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥})) → 𝐴 ≼ (𝐵 ∖ {𝑥})) |
44 | 28, 31, 42, 43 | syl3anc 1318 |
. . . . . . 7
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ≼ (𝐵 ∖ {𝑥})) |
45 | | eldifi 3694 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝑥 ∈ 𝐵) |
46 | 45 | ad2antll 761 |
. . . . . . . 8
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑥 ∈ 𝐵) |
47 | | simplr 788 |
. . . . . . . 8
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐶 ∈ 𝐵) |
48 | | difsnen 7927 |
. . . . . . . 8
⊢ ((𝐵 ∈ V ∧ 𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶})) |
49 | 29, 46, 47, 48 | syl3anc 1318 |
. . . . . . 7
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶})) |
50 | | domentr 7901 |
. . . . . . 7
⊢ ((𝐴 ≼ (𝐵 ∖ {𝑥}) ∧ (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶})) → 𝐴 ≼ (𝐵 ∖ {𝐶})) |
51 | 44, 49, 50 | syl2anc 691 |
. . . . . 6
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ≼ (𝐵 ∖ {𝐶})) |
52 | 51 | expr 641 |
. . . . 5
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝐴 ≼ (𝐵 ∖ {𝐶}))) |
53 | 52 | exlimdv 1848 |
. . . 4
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝐴 ≼ (𝐵 ∖ {𝐶}))) |
54 | 26, 53 | mpd 15 |
. . 3
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶})) |
55 | 7, 54 | exlimddv 1850 |
. 2
⊢ ((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶})) |
56 | 1 | adantr 480 |
. . 3
⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → 𝐴 ≼ 𝐵) |
57 | | difsn 4269 |
. . . . 5
⊢ (¬
𝐶 ∈ 𝐵 → (𝐵 ∖ {𝐶}) = 𝐵) |
58 | 57 | breq2d 4595 |
. . . 4
⊢ (¬
𝐶 ∈ 𝐵 → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴 ≼ 𝐵)) |
59 | 58 | adantl 481 |
. . 3
⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴 ≼ 𝐵)) |
60 | 56, 59 | mpbird 246 |
. 2
⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶})) |
61 | 55, 60 | pm2.61dan 828 |
1
⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ (𝐵 ∖ {𝐶})) |