| Step | Hyp | Ref
| Expression |
| 1 | | djueq2 9946 |
. . . . . . 7
⊢ (𝑥 = 𝐵 → (𝐴 ⊔ 𝑥) = (𝐴 ⊔ 𝐵)) |
| 2 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵)) |
| 3 | 1, 2 | breq12d 5156 |
. . . . . 6
⊢ (𝑥 = 𝐵 → ((𝐴 ⊔ 𝑥) ≈ (𝐴 +o 𝑥) ↔ (𝐴 ⊔ 𝐵) ≈ (𝐴 +o 𝐵))) |
| 4 | 3 | imbi2d 340 |
. . . . 5
⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ω → (𝐴 ⊔ 𝑥) ≈ (𝐴 +o 𝑥)) ↔ (𝐴 ∈ ω → (𝐴 ⊔ 𝐵) ≈ (𝐴 +o 𝐵)))) |
| 5 | | djueq2 9946 |
. . . . . . 7
⊢ (𝑥 = ∅ → (𝐴 ⊔ 𝑥) = (𝐴 ⊔ ∅)) |
| 6 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅)) |
| 7 | 5, 6 | breq12d 5156 |
. . . . . 6
⊢ (𝑥 = ∅ → ((𝐴 ⊔ 𝑥) ≈ (𝐴 +o 𝑥) ↔ (𝐴 ⊔ ∅) ≈ (𝐴 +o
∅))) |
| 8 | | djueq2 9946 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝐴 ⊔ 𝑥) = (𝐴 ⊔ 𝑦)) |
| 9 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦)) |
| 10 | 8, 9 | breq12d 5156 |
. . . . . 6
⊢ (𝑥 = 𝑦 → ((𝐴 ⊔ 𝑥) ≈ (𝐴 +o 𝑥) ↔ (𝐴 ⊔ 𝑦) ≈ (𝐴 +o 𝑦))) |
| 11 | | djueq2 9946 |
. . . . . . 7
⊢ (𝑥 = suc 𝑦 → (𝐴 ⊔ 𝑥) = (𝐴 ⊔ suc 𝑦)) |
| 12 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦)) |
| 13 | 11, 12 | breq12d 5156 |
. . . . . 6
⊢ (𝑥 = suc 𝑦 → ((𝐴 ⊔ 𝑥) ≈ (𝐴 +o 𝑥) ↔ (𝐴 ⊔ suc 𝑦) ≈ (𝐴 +o suc 𝑦))) |
| 14 | | dju0en 10216 |
. . . . . . 7
⊢ (𝐴 ∈ ω → (𝐴 ⊔ ∅) ≈ 𝐴) |
| 15 | | nna0 8642 |
. . . . . . 7
⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴) |
| 16 | 14, 15 | breqtrrd 5171 |
. . . . . 6
⊢ (𝐴 ∈ ω → (𝐴 ⊔ ∅) ≈
(𝐴 +o
∅)) |
| 17 | | 1oex 8516 |
. . . . . . . . . . 11
⊢
1o ∈ V |
| 18 | | djuassen 10219 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω ∧
1o ∈ V) → ((𝐴 ⊔ 𝑦) ⊔ 1o) ≈ (𝐴 ⊔ (𝑦 ⊔ 1o))) |
| 19 | 17, 18 | mp3an3 1452 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ⊔ 𝑦) ⊔ 1o) ≈ (𝐴 ⊔ (𝑦 ⊔ 1o))) |
| 20 | | enrefg 9024 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴) |
| 21 | | nnord 7895 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ω → Ord 𝑦) |
| 22 | | ordirr 6402 |
. . . . . . . . . . . . 13
⊢ (Ord
𝑦 → ¬ 𝑦 ∈ 𝑦) |
| 23 | 21, 22 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ω → ¬
𝑦 ∈ 𝑦) |
| 24 | | dju1en 10212 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ ω ∧ ¬
𝑦 ∈ 𝑦) → (𝑦 ⊔ 1o) ≈ suc 𝑦) |
| 25 | 23, 24 | mpdan 687 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ω → (𝑦 ⊔ 1o) ≈
suc 𝑦) |
| 26 | | djuen 10210 |
. . . . . . . . . . 11
⊢ ((𝐴 ≈ 𝐴 ∧ (𝑦 ⊔ 1o) ≈ suc 𝑦) → (𝐴 ⊔ (𝑦 ⊔ 1o)) ≈ (𝐴 ⊔ suc 𝑦)) |
| 27 | 20, 25, 26 | syl2an 596 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ⊔ (𝑦 ⊔ 1o)) ≈ (𝐴 ⊔ suc 𝑦)) |
| 28 | | entr 9046 |
. . . . . . . . . 10
⊢ ((((𝐴 ⊔ 𝑦) ⊔ 1o) ≈ (𝐴 ⊔ (𝑦 ⊔ 1o)) ∧ (𝐴 ⊔ (𝑦 ⊔ 1o)) ≈ (𝐴 ⊔ suc 𝑦)) → ((𝐴 ⊔ 𝑦) ⊔ 1o) ≈ (𝐴 ⊔ suc 𝑦)) |
| 29 | 19, 27, 28 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ⊔ 𝑦) ⊔ 1o) ≈ (𝐴 ⊔ suc 𝑦)) |
| 30 | 29 | ensymd 9045 |
. . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ⊔ suc 𝑦) ≈ ((𝐴 ⊔ 𝑦) ⊔ 1o)) |
| 31 | 17 | enref 9025 |
. . . . . . . . . . . 12
⊢
1o ≈ 1o |
| 32 | | djuen 10210 |
. . . . . . . . . . . 12
⊢ (((𝐴 ⊔ 𝑦) ≈ (𝐴 +o 𝑦) ∧ 1o ≈ 1o)
→ ((𝐴 ⊔ 𝑦) ⊔ 1o)
≈ ((𝐴 +o
𝑦) ⊔
1o)) |
| 33 | 31, 32 | mpan2 691 |
. . . . . . . . . . 11
⊢ ((𝐴 ⊔ 𝑦) ≈ (𝐴 +o 𝑦) → ((𝐴 ⊔ 𝑦) ⊔ 1o) ≈ ((𝐴 +o 𝑦) ⊔
1o)) |
| 34 | 33 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ⊔ 𝑦) ≈ (𝐴 +o 𝑦) → ((𝐴 ⊔ 𝑦) ⊔ 1o) ≈ ((𝐴 +o 𝑦) ⊔
1o))) |
| 35 | | nnacl 8649 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) ∈
ω) |
| 36 | | nnord 7895 |
. . . . . . . . . . . . 13
⊢ ((𝐴 +o 𝑦) ∈ ω → Ord
(𝐴 +o 𝑦)) |
| 37 | | ordirr 6402 |
. . . . . . . . . . . . 13
⊢ (Ord
(𝐴 +o 𝑦) → ¬ (𝐴 +o 𝑦) ∈ (𝐴 +o 𝑦)) |
| 38 | 35, 36, 37 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ¬
(𝐴 +o 𝑦) ∈ (𝐴 +o 𝑦)) |
| 39 | | dju1en 10212 |
. . . . . . . . . . . 12
⊢ (((𝐴 +o 𝑦) ∈ ω ∧ ¬
(𝐴 +o 𝑦) ∈ (𝐴 +o 𝑦)) → ((𝐴 +o 𝑦) ⊔ 1o) ≈ suc (𝐴 +o 𝑦)) |
| 40 | 35, 38, 39 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +o 𝑦) ⊔ 1o)
≈ suc (𝐴
+o 𝑦)) |
| 41 | | nnasuc 8644 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦)) |
| 42 | 40, 41 | breqtrrd 5171 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +o 𝑦) ⊔ 1o)
≈ (𝐴 +o
suc 𝑦)) |
| 43 | 34, 42 | jctird 526 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ⊔ 𝑦) ≈ (𝐴 +o 𝑦) → (((𝐴 ⊔ 𝑦) ⊔ 1o) ≈ ((𝐴 +o 𝑦) ⊔ 1o) ∧
((𝐴 +o 𝑦) ⊔ 1o)
≈ (𝐴 +o
suc 𝑦)))) |
| 44 | | entr 9046 |
. . . . . . . . 9
⊢ ((((𝐴 ⊔ 𝑦) ⊔ 1o) ≈ ((𝐴 +o 𝑦) ⊔ 1o) ∧
((𝐴 +o 𝑦) ⊔ 1o)
≈ (𝐴 +o
suc 𝑦)) → ((𝐴 ⊔ 𝑦) ⊔ 1o) ≈ (𝐴 +o suc 𝑦)) |
| 45 | 43, 44 | syl6 35 |
. . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ⊔ 𝑦) ≈ (𝐴 +o 𝑦) → ((𝐴 ⊔ 𝑦) ⊔ 1o) ≈ (𝐴 +o suc 𝑦))) |
| 46 | | entr 9046 |
. . . . . . . 8
⊢ (((𝐴 ⊔ suc 𝑦) ≈ ((𝐴 ⊔ 𝑦) ⊔ 1o) ∧ ((𝐴 ⊔ 𝑦) ⊔ 1o) ≈ (𝐴 +o suc 𝑦)) → (𝐴 ⊔ suc 𝑦) ≈ (𝐴 +o suc 𝑦)) |
| 47 | 30, 45, 46 | syl6an 684 |
. . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ⊔ 𝑦) ≈ (𝐴 +o 𝑦) → (𝐴 ⊔ suc 𝑦) ≈ (𝐴 +o suc 𝑦))) |
| 48 | 47 | expcom 413 |
. . . . . 6
⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 ⊔ 𝑦) ≈ (𝐴 +o 𝑦) → (𝐴 ⊔ suc 𝑦) ≈ (𝐴 +o suc 𝑦)))) |
| 49 | 7, 10, 13, 16, 48 | finds2 7920 |
. . . . 5
⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (𝐴 ⊔ 𝑥) ≈ (𝐴 +o 𝑥))) |
| 50 | 4, 49 | vtoclga 3577 |
. . . 4
⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ⊔ 𝐵) ≈ (𝐴 +o 𝐵))) |
| 51 | 50 | impcom 407 |
. . 3
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊔ 𝐵) ≈ (𝐴 +o 𝐵)) |
| 52 | | carden2b 10007 |
. . 3
⊢ ((𝐴 ⊔ 𝐵) ≈ (𝐴 +o 𝐵) → (card‘(𝐴 ⊔ 𝐵)) = (card‘(𝐴 +o 𝐵))) |
| 53 | 51, 52 | syl 17 |
. 2
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) →
(card‘(𝐴 ⊔
𝐵)) = (card‘(𝐴 +o 𝐵))) |
| 54 | | nnacl 8649 |
. . 3
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω) |
| 55 | | cardnn 10003 |
. . 3
⊢ ((𝐴 +o 𝐵) ∈ ω → (card‘(𝐴 +o 𝐵)) = (𝐴 +o 𝐵)) |
| 56 | 54, 55 | syl 17 |
. 2
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) →
(card‘(𝐴
+o 𝐵)) = (𝐴 +o 𝐵)) |
| 57 | 53, 56 | eqtrd 2777 |
1
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) →
(card‘(𝐴 ⊔
𝐵)) = (𝐴 +o 𝐵)) |