| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | sdomdom 9021 | . . 3
⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | 
| 2 |  | brdomi 9000 | . . 3
⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) | 
| 3 | 1, 2 | syl 17 | . 2
⊢ (𝐴 ≺ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) | 
| 4 |  | relsdom 8993 | . . . . . . 7
⊢ Rel
≺ | 
| 5 | 4 | brrelex1i 5740 | . . . . . 6
⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V) | 
| 6 | 5 | adantr 480 | . . . . 5
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ∈ V) | 
| 7 |  | vex 3483 | . . . . . . 7
⊢ 𝑓 ∈ V | 
| 8 | 7 | rnex 7933 | . . . . . 6
⊢ ran 𝑓 ∈ V | 
| 9 | 8 | a1i 11 | . . . . 5
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ∈ V) | 
| 10 |  | f1f1orn 6858 | . . . . . . 7
⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴–1-1-onto→ran
𝑓) | 
| 11 | 10 | adantl 481 | . . . . . 6
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝑓:𝐴–1-1-onto→ran
𝑓) | 
| 12 |  | f1of1 6846 | . . . . . 6
⊢ (𝑓:𝐴–1-1-onto→ran
𝑓 → 𝑓:𝐴–1-1→ran 𝑓) | 
| 13 | 11, 12 | syl 17 | . . . . 5
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝑓:𝐴–1-1→ran 𝑓) | 
| 14 |  | f1dom2g 9011 | . . . . 5
⊢ ((𝐴 ∈ V ∧ ran 𝑓 ∈ V ∧ 𝑓:𝐴–1-1→ran 𝑓) → 𝐴 ≼ ran 𝑓) | 
| 15 | 6, 9, 13, 14 | syl3anc 1372 | . . . 4
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ≼ ran 𝑓) | 
| 16 |  | sdomnen 9022 | . . . . . . . 8
⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | 
| 17 | 16 | adantr 480 | . . . . . . 7
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ¬ 𝐴 ≈ 𝐵) | 
| 18 |  | ssdif0 4365 | . . . . . . . 8
⊢ (𝐵 ⊆ ran 𝑓 ↔ (𝐵 ∖ ran 𝑓) = ∅) | 
| 19 |  | simplr 768 | . . . . . . . . . . 11
⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴–1-1→𝐵) | 
| 20 |  | f1f 6803 | . . . . . . . . . . . . . 14
⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) | 
| 21 | 20 | frnd 6743 | . . . . . . . . . . . . 13
⊢ (𝑓:𝐴–1-1→𝐵 → ran 𝑓 ⊆ 𝐵) | 
| 22 | 19, 21 | syl 17 | . . . . . . . . . . . 12
⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓 ⊆ 𝐵) | 
| 23 |  | simpr 484 | . . . . . . . . . . . 12
⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐵 ⊆ ran 𝑓) | 
| 24 | 22, 23 | eqssd 4000 | . . . . . . . . . . 11
⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓 = 𝐵) | 
| 25 |  | dff1o5 6856 | . . . . . . . . . . 11
⊢ (𝑓:𝐴–1-1-onto→𝐵 ↔ (𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵)) | 
| 26 | 19, 24, 25 | sylanbrc 583 | . . . . . . . . . 10
⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴–1-1-onto→𝐵) | 
| 27 |  | f1oen3g 9008 | . . . . . . . . . 10
⊢ ((𝑓 ∈ V ∧ 𝑓:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | 
| 28 | 7, 26, 27 | sylancr 587 | . . . . . . . . 9
⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐴 ≈ 𝐵) | 
| 29 | 28 | ex 412 | . . . . . . . 8
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐵 ⊆ ran 𝑓 → 𝐴 ≈ 𝐵)) | 
| 30 | 18, 29 | biimtrrid 243 | . . . . . . 7
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ((𝐵 ∖ ran 𝑓) = ∅ → 𝐴 ≈ 𝐵)) | 
| 31 | 17, 30 | mtod 198 | . . . . . 6
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ¬ (𝐵 ∖ ran 𝑓) = ∅) | 
| 32 |  | neq0 4351 | . . . . . 6
⊢ (¬
(𝐵 ∖ ran 𝑓) = ∅ ↔ ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓)) | 
| 33 | 31, 32 | sylib 218 | . . . . 5
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓)) | 
| 34 |  | snssi 4807 | . . . . . . 7
⊢ (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝑤} ⊆ (𝐵 ∖ ran 𝑓)) | 
| 35 |  | vex 3483 | . . . . . . . . 9
⊢ 𝑤 ∈ V | 
| 36 |  | en2sn 9082 | . . . . . . . . 9
⊢ ((𝐴 ∈ V ∧ 𝑤 ∈ V) → {𝐴} ≈ {𝑤}) | 
| 37 | 6, 35, 36 | sylancl 586 | . . . . . . . 8
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → {𝐴} ≈ {𝑤}) | 
| 38 | 4 | brrelex2i 5741 | . . . . . . . . . 10
⊢ (𝐴 ≺ 𝐵 → 𝐵 ∈ V) | 
| 39 | 38 | adantr 480 | . . . . . . . . 9
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐵 ∈ V) | 
| 40 |  | difexg 5328 | . . . . . . . . 9
⊢ (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V) | 
| 41 |  | ssdomg 9041 | . . . . . . . . 9
⊢ ((𝐵 ∖ ran 𝑓) ∈ V → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝑤} ≼ (𝐵 ∖ ran 𝑓))) | 
| 42 | 39, 40, 41 | 3syl 18 | . . . . . . . 8
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝑤} ≼ (𝐵 ∖ ran 𝑓))) | 
| 43 |  | endomtr 9053 | . . . . . . . 8
⊢ (({𝐴} ≈ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)) | 
| 44 | 37, 42, 43 | syl6an 684 | . . . . . . 7
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))) | 
| 45 | 34, 44 | syl5 34 | . . . . . 6
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))) | 
| 46 | 45 | exlimdv 1932 | . . . . 5
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))) | 
| 47 | 33, 46 | mpd 15 | . . . 4
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)) | 
| 48 |  | disjdif 4471 | . . . . 5
⊢ (ran
𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅ | 
| 49 | 48 | a1i 11 | . . . 4
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅) | 
| 50 |  | undom 9100 | . . . 4
⊢ (((𝐴 ≼ ran 𝑓 ∧ {𝐴} ≼ (𝐵 ∖ ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓))) | 
| 51 | 15, 47, 49, 50 | syl21anc 837 | . . 3
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓))) | 
| 52 |  | df-suc 6389 | . . . 4
⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | 
| 53 | 52 | a1i 11 | . . 3
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → suc 𝐴 = (𝐴 ∪ {𝐴})) | 
| 54 |  | undif2 4476 | . . . 4
⊢ (ran
𝑓 ∪ (𝐵 ∖ ran 𝑓)) = (ran 𝑓 ∪ 𝐵) | 
| 55 | 21 | adantl 481 | . . . . 5
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ⊆ 𝐵) | 
| 56 |  | ssequn1 4185 | . . . . 5
⊢ (ran
𝑓 ⊆ 𝐵 ↔ (ran 𝑓 ∪ 𝐵) = 𝐵) | 
| 57 | 55, 56 | sylib 218 | . . . 4
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (ran 𝑓 ∪ 𝐵) = 𝐵) | 
| 58 | 54, 57 | eqtr2id 2789 | . . 3
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐵 = (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓))) | 
| 59 | 51, 53, 58 | 3brtr4d 5174 | . 2
⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → suc 𝐴 ≼ 𝐵) | 
| 60 | 3, 59 | exlimddv 1934 | 1
⊢ (𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵) |