| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | sdomdom 9020 | . . 3
⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | 
| 2 |  | brdomi 8999 | . . 3
⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) | 
| 3 | 1, 2 | syl 17 | . 2
⊢ (𝐴 ≺ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) | 
| 4 |  | vex 3484 | . . . . 5
⊢ 𝑓 ∈ V | 
| 5 | 4 | rnex 7932 | . . . . 5
⊢ ran 𝑓 ∈ V | 
| 6 |  | f1f1orn 6859 | . . . . . . 7
⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴–1-1-onto→ran
𝑓) | 
| 7 | 6 | adantl 481 | . . . . . 6
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝑓:𝐴–1-1-onto→ran
𝑓) | 
| 8 |  | f1of1 6847 | . . . . . 6
⊢ (𝑓:𝐴–1-1-onto→ran
𝑓 → 𝑓:𝐴–1-1→ran 𝑓) | 
| 9 | 7, 8 | syl 17 | . . . . 5
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝑓:𝐴–1-1→ran 𝑓) | 
| 10 |  | f1dom3g 9008 | . . . . 5
⊢ ((𝑓 ∈ V ∧ ran 𝑓 ∈ V ∧ 𝑓:𝐴–1-1→ran 𝑓) → 𝐴 ≼ ran 𝑓) | 
| 11 | 4, 5, 9, 10 | mp3an12i 1467 | . . . 4
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ≼ ran 𝑓) | 
| 12 |  | sdomnen 9021 | . . . . . . . 8
⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | 
| 13 | 12 | adantr 480 | . . . . . . 7
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ¬ 𝐴 ≈ 𝐵) | 
| 14 |  | ssdif0 4366 | . . . . . . . 8
⊢ (𝐵 ⊆ ran 𝑓 ↔ (𝐵 ∖ ran 𝑓) = ∅) | 
| 15 |  | simplr 769 | . . . . . . . . . . 11
⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴–1-1→𝐵) | 
| 16 |  | f1f 6804 | . . . . . . . . . . . . . 14
⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) | 
| 17 | 16 | frnd 6744 | . . . . . . . . . . . . 13
⊢ (𝑓:𝐴–1-1→𝐵 → ran 𝑓 ⊆ 𝐵) | 
| 18 | 15, 17 | syl 17 | . . . . . . . . . . . 12
⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓 ⊆ 𝐵) | 
| 19 |  | simpr 484 | . . . . . . . . . . . 12
⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐵 ⊆ ran 𝑓) | 
| 20 | 18, 19 | eqssd 4001 | . . . . . . . . . . 11
⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓 = 𝐵) | 
| 21 |  | dff1o5 6857 | . . . . . . . . . . 11
⊢ (𝑓:𝐴–1-1-onto→𝐵 ↔ (𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵)) | 
| 22 | 15, 20, 21 | sylanbrc 583 | . . . . . . . . . 10
⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴–1-1-onto→𝐵) | 
| 23 |  | f1oen3g 9007 | . . . . . . . . . 10
⊢ ((𝑓 ∈ V ∧ 𝑓:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | 
| 24 | 4, 22, 23 | sylancr 587 | . . . . . . . . 9
⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐴 ≈ 𝐵) | 
| 25 | 24 | ex 412 | . . . . . . . 8
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐵 ⊆ ran 𝑓 → 𝐴 ≈ 𝐵)) | 
| 26 | 14, 25 | biimtrrid 243 | . . . . . . 7
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ((𝐵 ∖ ran 𝑓) = ∅ → 𝐴 ≈ 𝐵)) | 
| 27 | 13, 26 | mtod 198 | . . . . . 6
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ¬ (𝐵 ∖ ran 𝑓) = ∅) | 
| 28 |  | neq0 4352 | . . . . . 6
⊢ (¬
(𝐵 ∖ ran 𝑓) = ∅ ↔ ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓)) | 
| 29 | 27, 28 | sylib 218 | . . . . 5
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓)) | 
| 30 |  | snssi 4808 | . . . . . . 7
⊢ (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝑤} ⊆ (𝐵 ∖ ran 𝑓)) | 
| 31 |  | relsdom 8992 | . . . . . . . . . . 11
⊢ Rel
≺ | 
| 32 | 31 | brrelex1i 5741 | . . . . . . . . . 10
⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V) | 
| 33 | 32 | adantr 480 | . . . . . . . . 9
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ∈ V) | 
| 34 |  | vex 3484 | . . . . . . . . 9
⊢ 𝑤 ∈ V | 
| 35 |  | en2sn 9081 | . . . . . . . . 9
⊢ ((𝐴 ∈ V ∧ 𝑤 ∈ V) → {𝐴} ≈ {𝑤}) | 
| 36 | 33, 34, 35 | sylancl 586 | . . . . . . . 8
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → {𝐴} ≈ {𝑤}) | 
| 37 | 31 | brrelex2i 5742 | . . . . . . . . . 10
⊢ (𝐴 ≺ 𝐵 → 𝐵 ∈ V) | 
| 38 | 37 | adantr 480 | . . . . . . . . 9
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐵 ∈ V) | 
| 39 |  | difexg 5329 | . . . . . . . . 9
⊢ (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V) | 
| 40 |  | snfi 9083 | . . . . . . . . . . 11
⊢ {𝑤} ∈ Fin | 
| 41 |  | ssdomfi2 9237 | . . . . . . . . . . 11
⊢ (({𝑤} ∈ Fin ∧ (𝐵 ∖ ran 𝑓) ∈ V ∧ {𝑤} ⊆ (𝐵 ∖ ran 𝑓)) → {𝑤} ≼ (𝐵 ∖ ran 𝑓)) | 
| 42 | 40, 41 | mp3an1 1450 | . . . . . . . . . 10
⊢ (((𝐵 ∖ ran 𝑓) ∈ V ∧ {𝑤} ⊆ (𝐵 ∖ ran 𝑓)) → {𝑤} ≼ (𝐵 ∖ ran 𝑓)) | 
| 43 | 42 | ex 412 | . . . . . . . . 9
⊢ ((𝐵 ∖ ran 𝑓) ∈ V → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝑤} ≼ (𝐵 ∖ ran 𝑓))) | 
| 44 | 38, 39, 43 | 3syl 18 | . . . . . . . 8
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝑤} ≼ (𝐵 ∖ ran 𝑓))) | 
| 45 |  | endom 9019 | . . . . . . . . 9
⊢ ({𝐴} ≈ {𝑤} → {𝐴} ≼ {𝑤}) | 
| 46 |  | domtrfi 9233 | . . . . . . . . . 10
⊢ (({𝑤} ∈ Fin ∧ {𝐴} ≼ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)) | 
| 47 | 40, 46 | mp3an1 1450 | . . . . . . . . 9
⊢ (({𝐴} ≼ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)) | 
| 48 | 45, 47 | sylan 580 | . . . . . . . 8
⊢ (({𝐴} ≈ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)) | 
| 49 | 36, 44, 48 | syl6an 684 | . . . . . . 7
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))) | 
| 50 | 30, 49 | syl5 34 | . . . . . 6
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))) | 
| 51 | 50 | exlimdv 1933 | . . . . 5
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))) | 
| 52 | 29, 51 | mpd 15 | . . . 4
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)) | 
| 53 |  | disjdif 4472 | . . . . 5
⊢ (ran
𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅ | 
| 54 | 53 | a1i 11 | . . . 4
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅) | 
| 55 |  | undom 9099 | . . . 4
⊢ (((𝐴 ≼ ran 𝑓 ∧ {𝐴} ≼ (𝐵 ∖ ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓))) | 
| 56 | 11, 52, 54, 55 | syl21anc 838 | . . 3
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓))) | 
| 57 |  | df-suc 6390 | . . . 4
⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | 
| 58 | 57 | a1i 11 | . . 3
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → suc 𝐴 = (𝐴 ∪ {𝐴})) | 
| 59 |  | undif2 4477 | . . . 4
⊢ (ran
𝑓 ∪ (𝐵 ∖ ran 𝑓)) = (ran 𝑓 ∪ 𝐵) | 
| 60 | 17 | adantl 481 | . . . . 5
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ⊆ 𝐵) | 
| 61 |  | ssequn1 4186 | . . . . 5
⊢ (ran
𝑓 ⊆ 𝐵 ↔ (ran 𝑓 ∪ 𝐵) = 𝐵) | 
| 62 | 60, 61 | sylib 218 | . . . 4
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (ran 𝑓 ∪ 𝐵) = 𝐵) | 
| 63 | 59, 62 | eqtr2id 2790 | . . 3
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐵 = (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓))) | 
| 64 | 56, 58, 63 | 3brtr4d 5175 | . 2
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → suc 𝐴 ≼ 𝐵) | 
| 65 | 3, 64 | exlimddv 1935 | 1
⊢ (𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵) |