Step | Hyp | Ref
| Expression |
1 | | sdomdom 8920 |
. . . . 5
⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) |
2 | | relsdom 8890 |
. . . . . . 7
⊢ Rel
≺ |
3 | 2 | brrelex2i 5689 |
. . . . . 6
⊢ (𝐴 ≺ 𝐵 → 𝐵 ∈ V) |
4 | | brdomg 8896 |
. . . . . 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 231 |
. . . 4
⊢ (𝐴 ≺ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) |
7 | 6 | adantr 481 |
. . 3
⊢ ((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) → ∃𝑓 𝑓:𝐴–1-1→𝐵) |
8 | | f1f 6738 |
. . . . . . . 8
⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) |
9 | 8 | frnd 6676 |
. . . . . . 7
⊢ (𝑓:𝐴–1-1→𝐵 → ran 𝑓 ⊆ 𝐵) |
10 | 9 | adantl 482 |
. . . . . 6
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ⊆ 𝐵) |
11 | | sdomnen 8921 |
. . . . . . . 8
⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) |
12 | 11 | ad2antrr 724 |
. . . . . . 7
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ¬ 𝐴 ≈ 𝐵) |
13 | | vex 3449 |
. . . . . . . . . . 11
⊢ 𝑓 ∈ V |
14 | | dff1o5 6793 |
. . . . . . . . . . . 12
⊢ (𝑓:𝐴–1-1-onto→𝐵 ↔ (𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵)) |
15 | 14 | biimpri 227 |
. . . . . . . . . . 11
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵) → 𝑓:𝐴–1-1-onto→𝐵) |
16 | | f1oen3g 8906 |
. . . . . . . . . . 11
⊢ ((𝑓 ∈ V ∧ 𝑓:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) |
17 | 13, 15, 16 | sylancr 587 |
. . . . . . . . . 10
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵) → 𝐴 ≈ 𝐵) |
18 | 17 | ex 413 |
. . . . . . . . 9
⊢ (𝑓:𝐴–1-1→𝐵 → (ran 𝑓 = 𝐵 → 𝐴 ≈ 𝐵)) |
19 | 18 | necon3bd 2957 |
. . . . . . . 8
⊢ (𝑓:𝐴–1-1→𝐵 → (¬ 𝐴 ≈ 𝐵 → ran 𝑓 ≠ 𝐵)) |
20 | 19 | adantl 482 |
. . . . . . 7
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (¬ 𝐴 ≈ 𝐵 → ran 𝑓 ≠ 𝐵)) |
21 | 12, 20 | mpd 15 |
. . . . . 6
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ≠ 𝐵) |
22 | | pssdifn0 4325 |
. . . . . 6
⊢ ((ran
𝑓 ⊆ 𝐵 ∧ ran 𝑓 ≠ 𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅) |
23 | 10, 21, 22 | syl2anc 584 |
. . . . 5
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅) |
24 | | n0 4306 |
. . . . 5
⊢ ((𝐵 ∖ ran 𝑓) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓)) |
25 | 23, 24 | sylib 217 |
. . . 4
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓)) |
26 | 2 | brrelex1i 5688 |
. . . . . . . . 9
⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V) |
27 | 26 | ad2antrr 724 |
. . . . . . . 8
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ∈ V) |
28 | 3 | ad2antrr 724 |
. . . . . . . . 9
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐵 ∈ V) |
29 | 28 | difexd 5286 |
. . . . . . . 8
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ∈ V) |
30 | | eldifn 4087 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → ¬ 𝑥 ∈ ran 𝑓) |
31 | | disjsn 4672 |
. . . . . . . . . . . . 13
⊢ ((ran
𝑓 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝑓) |
32 | 30, 31 | sylibr 233 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → (ran 𝑓 ∩ {𝑥}) = ∅) |
33 | 32 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → (ran 𝑓 ∩ {𝑥}) = ∅) |
34 | 9 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ran 𝑓 ⊆ 𝐵) |
35 | | reldisj 4411 |
. . . . . . . . . . . 12
⊢ (ran
𝑓 ⊆ 𝐵 → ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ran 𝑓 ⊆ (𝐵 ∖ {𝑥}))) |
36 | 34, 35 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ran 𝑓 ⊆ (𝐵 ∖ {𝑥}))) |
37 | 33, 36 | mpbid 231 |
. . . . . . . . . 10
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ran 𝑓 ⊆ (𝐵 ∖ {𝑥})) |
38 | | f1ssr 6745 |
. . . . . . . . . 10
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})) → 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥})) |
39 | 37, 38 | syldan 591 |
. . . . . . . . 9
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥})) |
40 | 39 | adantl 482 |
. . . . . . . 8
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥})) |
41 | | f1dom2g 8909 |
. . . . . . . 8
⊢ ((𝐴 ∈ V ∧ (𝐵 ∖ {𝑥}) ∈ V ∧ 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥})) → 𝐴 ≼ (𝐵 ∖ {𝑥})) |
42 | 27, 29, 40, 41 | syl3anc 1371 |
. . . . . . 7
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ≼ (𝐵 ∖ {𝑥})) |
43 | | eldifi 4086 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝑥 ∈ 𝐵) |
44 | 43 | ad2antll 727 |
. . . . . . . 8
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑥 ∈ 𝐵) |
45 | | simplr 767 |
. . . . . . . 8
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐶 ∈ 𝐵) |
46 | | difsnen 8997 |
. . . . . . . 8
⊢ ((𝐵 ∈ V ∧ 𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶})) |
47 | 28, 44, 45, 46 | syl3anc 1371 |
. . . . . . 7
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶})) |
48 | | domentr 8953 |
. . . . . . 7
⊢ ((𝐴 ≼ (𝐵 ∖ {𝑥}) ∧ (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶})) → 𝐴 ≼ (𝐵 ∖ {𝐶})) |
49 | 42, 47, 48 | syl2anc 584 |
. . . . . 6
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ≼ (𝐵 ∖ {𝐶})) |
50 | 49 | expr 457 |
. . . . 5
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝐴 ≼ (𝐵 ∖ {𝐶}))) |
51 | 50 | exlimdv 1936 |
. . . 4
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝐴 ≼ (𝐵 ∖ {𝐶}))) |
52 | 25, 51 | mpd 15 |
. . 3
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶})) |
53 | 7, 52 | exlimddv 1938 |
. 2
⊢ ((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶})) |
54 | 1 | adantr 481 |
. . 3
⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → 𝐴 ≼ 𝐵) |
55 | | difsn 4758 |
. . . . 5
⊢ (¬
𝐶 ∈ 𝐵 → (𝐵 ∖ {𝐶}) = 𝐵) |
56 | 55 | breq2d 5117 |
. . . 4
⊢ (¬
𝐶 ∈ 𝐵 → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴 ≼ 𝐵)) |
57 | 56 | adantl 482 |
. . 3
⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴 ≼ 𝐵)) |
58 | 54, 57 | mpbird 256 |
. 2
⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶})) |
59 | 53, 58 | pm2.61dan 811 |
1
⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ (𝐵 ∖ {𝐶})) |