Proof of Theorem canthnumlem
| Step | Hyp | Ref
| Expression |
| 1 | | f1f 6804 |
. . . . 5
⊢ (𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴 → 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴) |
| 2 | | ssid 4006 |
. . . . . 6
⊢
(𝒫 𝐴 ∩
dom card) ⊆ (𝒫 𝐴 ∩ dom card) |
| 3 | | canth4.1 |
. . . . . . 7
⊢ 𝑊 = {〈𝑥, 𝑟〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))} |
| 4 | | canth4.2 |
. . . . . . 7
⊢ 𝐵 = ∪
dom 𝑊 |
| 5 | | canth4.3 |
. . . . . . 7
⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}) |
| 6 | 3, 4, 5 | canth4 10687 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ (𝒫 𝐴 ∩ dom card)) → (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ (𝐹‘𝐵) = (𝐹‘𝐶))) |
| 7 | 2, 6 | mp3an3 1452 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ (𝐹‘𝐵) = (𝐹‘𝐶))) |
| 8 | 1, 7 | sylan2 593 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ (𝐹‘𝐵) = (𝐹‘𝐶))) |
| 9 | 8 | simp3d 1145 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐹‘𝐵) = (𝐹‘𝐶)) |
| 10 | | simpr 484 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) |
| 11 | 8 | simp1d 1143 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ⊆ 𝐴) |
| 12 | | elpw2g 5333 |
. . . . . . 7
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴)) |
| 13 | 12 | adantr 480 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴)) |
| 14 | 11, 13 | mpbird 257 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ∈ 𝒫 𝐴) |
| 15 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ 𝐵 = 𝐵 |
| 16 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (𝑊‘𝐵) = (𝑊‘𝐵) |
| 17 | 15, 16 | pm3.2i 470 |
. . . . . . . . . . 11
⊢ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵)) |
| 18 | | simpl 482 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐴 ∈ 𝑉) |
| 19 | 10, 1 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴) |
| 20 | 19 | ffvelcdmda 7104 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹‘𝑥) ∈ 𝐴) |
| 21 | 3, 18, 20, 4 | fpwwe 10686 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ((𝐵𝑊(𝑊‘𝐵) ∧ (𝐹‘𝐵) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵)))) |
| 22 | 17, 21 | mpbiri 258 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐵𝑊(𝑊‘𝐵) ∧ (𝐹‘𝐵) ∈ 𝐵)) |
| 23 | 22 | simpld 494 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵𝑊(𝑊‘𝐵)) |
| 24 | 3, 18 | fpwwelem 10685 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐵𝑊(𝑊‘𝐵) ↔ ((𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦)))) |
| 25 | 23, 24 | mpbid 232 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ((𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦))) |
| 26 | 25 | simprld 772 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝑊‘𝐵) We 𝐵) |
| 27 | | fvex 6919 |
. . . . . . . 8
⊢ (𝑊‘𝐵) ∈ V |
| 28 | | weeq1 5672 |
. . . . . . . 8
⊢ (𝑟 = (𝑊‘𝐵) → (𝑟 We 𝐵 ↔ (𝑊‘𝐵) We 𝐵)) |
| 29 | 27, 28 | spcev 3606 |
. . . . . . 7
⊢ ((𝑊‘𝐵) We 𝐵 → ∃𝑟 𝑟 We 𝐵) |
| 30 | 26, 29 | syl 17 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ∃𝑟 𝑟 We 𝐵) |
| 31 | | ween 10075 |
. . . . . 6
⊢ (𝐵 ∈ dom card ↔
∃𝑟 𝑟 We 𝐵) |
| 32 | 30, 31 | sylibr 234 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ∈ dom card) |
| 33 | 14, 32 | elind 4200 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ∈ (𝒫 𝐴 ∩ dom card)) |
| 34 | 8 | simp2d 1144 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ⊊ 𝐵) |
| 35 | 34 | pssssd 4100 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ⊆ 𝐵) |
| 36 | 35, 11 | sstrd 3994 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ⊆ 𝐴) |
| 37 | | elpw2g 5333 |
. . . . . . 7
⊢ (𝐴 ∈ 𝑉 → (𝐶 ∈ 𝒫 𝐴 ↔ 𝐶 ⊆ 𝐴)) |
| 38 | 37 | adantr 480 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐶 ∈ 𝒫 𝐴 ↔ 𝐶 ⊆ 𝐴)) |
| 39 | 36, 38 | mpbird 257 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ∈ 𝒫 𝐴) |
| 40 | | ssnum 10079 |
. . . . . 6
⊢ ((𝐵 ∈ dom card ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ dom card) |
| 41 | 32, 35, 40 | syl2anc 584 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ∈ dom card) |
| 42 | 39, 41 | elind 4200 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ∈ (𝒫 𝐴 ∩ dom card)) |
| 43 | | f1fveq 7282 |
. . . 4
⊢ ((𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴 ∧ (𝐵 ∈ (𝒫 𝐴 ∩ dom card) ∧ 𝐶 ∈ (𝒫 𝐴 ∩ dom card))) → ((𝐹‘𝐵) = (𝐹‘𝐶) ↔ 𝐵 = 𝐶)) |
| 44 | 10, 33, 42, 43 | syl12anc 837 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ((𝐹‘𝐵) = (𝐹‘𝐶) ↔ 𝐵 = 𝐶)) |
| 45 | 9, 44 | mpbid 232 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 = 𝐶) |
| 46 | 34 | pssned 4101 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ≠ 𝐵) |
| 47 | 46 | necomd 2996 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ≠ 𝐶) |
| 48 | 47 | neneqd 2945 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ¬ 𝐵 = 𝐶) |
| 49 | 45, 48 | pm2.65da 817 |
1
⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) |