| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | sdomdom 9021 | . . . . 5
⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | 
| 2 |  | relsdom 8993 | . . . . . . 7
⊢ Rel
≺ | 
| 3 | 2 | brrelex2i 5741 | . . . . . 6
⊢ (𝐴 ≺ 𝐵 → 𝐵 ∈ V) | 
| 4 |  | brdomg 8998 | . . . . . 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 232 | . . . 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 6803 | . . . . . . . 8
⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) | 
| 9 | 8 | frnd 6743 | . . . . . . 7
⊢ (𝑓:𝐴–1-1→𝐵 → ran 𝑓 ⊆ 𝐵) | 
| 10 | 9 | adantl 481 | . . . . . 6
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ⊆ 𝐵) | 
| 11 |  | sdomnen 9022 | . . . . . . . 8
⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | 
| 12 | 11 | ad2antrr 726 | . . . . . . 7
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ¬ 𝐴 ≈ 𝐵) | 
| 13 |  | vex 3483 | . . . . . . . . . . 11
⊢ 𝑓 ∈ V | 
| 14 |  | dff1o5 6856 | . . . . . . . . . . . 12
⊢ (𝑓:𝐴–1-1-onto→𝐵 ↔ (𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵)) | 
| 15 | 14 | biimpri 228 | . . . . . . . . . . 11
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵) → 𝑓:𝐴–1-1-onto→𝐵) | 
| 16 |  | f1oen3g 9008 | . . . . . . . . . . 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 412 | . . . . . . . . 9
⊢ (𝑓:𝐴–1-1→𝐵 → (ran 𝑓 = 𝐵 → 𝐴 ≈ 𝐵)) | 
| 19 | 18 | necon3bd 2953 | . . . . . . . 8
⊢ (𝑓:𝐴–1-1→𝐵 → (¬ 𝐴 ≈ 𝐵 → ran 𝑓 ≠ 𝐵)) | 
| 20 | 19 | adantl 481 | . . . . . . 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 4367 | . . . . . 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 4352 | . . . . 5
⊢ ((𝐵 ∖ ran 𝑓) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓)) | 
| 25 | 23, 24 | sylib 218 | . . . 4
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓)) | 
| 26 | 2 | brrelex1i 5740 | . . . . . . . . 9
⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V) | 
| 27 | 26 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ∈ V) | 
| 28 | 3 | ad2antrr 726 | . . . . . . . . 9
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐵 ∈ V) | 
| 29 | 28 | difexd 5330 | . . . . . . . 8
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ∈ V) | 
| 30 |  | eldifn 4131 | . . . . . . . . . . . . 13
⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → ¬ 𝑥 ∈ ran 𝑓) | 
| 31 |  | disjsn 4710 | . . . . . . . . . . . . 13
⊢ ((ran
𝑓 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝑓) | 
| 32 | 30, 31 | sylibr 234 | . . . . . . . . . . . 12
⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → (ran 𝑓 ∩ {𝑥}) = ∅) | 
| 33 | 32 | adantl 481 | . . . . . . . . . . 11
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → (ran 𝑓 ∩ {𝑥}) = ∅) | 
| 34 | 9 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ran 𝑓 ⊆ 𝐵) | 
| 35 |  | reldisj 4452 | . . . . . . . . . . . 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 232 | . . . . . . . . . 10
⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ran 𝑓 ⊆ (𝐵 ∖ {𝑥})) | 
| 38 |  | f1ssr 6809 | . . . . . . . . . 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 481 | . . . . . . . 8
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥})) | 
| 41 |  | f1dom2g 9011 | . . . . . . . 8
⊢ ((𝐴 ∈ V ∧ (𝐵 ∖ {𝑥}) ∈ V ∧ 𝑓:𝐴–1-1→(𝐵 ∖ {𝑥})) → 𝐴 ≼ (𝐵 ∖ {𝑥})) | 
| 42 | 27, 29, 40, 41 | syl3anc 1372 | . . . . . . 7
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ≼ (𝐵 ∖ {𝑥})) | 
| 43 |  | eldifi 4130 | . . . . . . . . 9
⊢ (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝑥 ∈ 𝐵) | 
| 44 | 43 | ad2antll 729 | . . . . . . . 8
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑥 ∈ 𝐵) | 
| 45 |  | simplr 768 | . . . . . . . 8
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐶 ∈ 𝐵) | 
| 46 |  | difsnen 9094 | . . . . . . . 8
⊢ ((𝐵 ∈ V ∧ 𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶})) | 
| 47 | 28, 44, 45, 46 | syl3anc 1372 | . . . . . . 7
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶})) | 
| 48 |  | domentr 9054 | . . . . . . 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 456 | . . . . 5
⊢ (((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑓:𝐴–1-1→𝐵) → (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝐴 ≼ (𝐵 ∖ {𝐶}))) | 
| 51 | 50 | exlimdv 1932 | . . . 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 1934 | . 2
⊢ ((𝐴 ≺ 𝐵 ∧ 𝐶 ∈ 𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶})) | 
| 54 | 1 | adantr 480 | . . 3
⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → 𝐴 ≼ 𝐵) | 
| 55 |  | difsn 4797 | . . . . 5
⊢ (¬
𝐶 ∈ 𝐵 → (𝐵 ∖ {𝐶}) = 𝐵) | 
| 56 | 55 | breq2d 5154 | . . . 4
⊢ (¬
𝐶 ∈ 𝐵 → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴 ≼ 𝐵)) | 
| 57 | 56 | adantl 481 | . . 3
⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴 ≼ 𝐵)) | 
| 58 | 54, 57 | mpbird 257 | . 2
⊢ ((𝐴 ≺ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶})) | 
| 59 | 53, 58 | pm2.61dan 812 | 1
⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ (𝐵 ∖ {𝐶})) |