Proof of Theorem oaomoencom
| Step | Hyp | Ref
| Expression |
| 1 | | oancom 9691 |
. . . 4
⊢
(1o +o ω) ≠ (ω +o
1o) |
| 2 | 1 | neii 2942 |
. . 3
⊢ ¬
(1o +o ω) = (ω +o
1o) |
| 3 | | 1on 8518 |
. . . 4
⊢
1o ∈ On |
| 4 | | omelon 9686 |
. . . . 5
⊢ ω
∈ On |
| 5 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑏 = ω → (1o
+o 𝑏) =
(1o +o ω)) |
| 6 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑏 = ω → (𝑏 +o 1o) =
(ω +o 1o)) |
| 7 | 5, 6 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑏 = ω →
((1o +o 𝑏) = (𝑏 +o 1o) ↔
(1o +o ω) = (ω +o
1o))) |
| 8 | 7 | notbid 318 |
. . . . . 6
⊢ (𝑏 = ω → (¬
(1o +o 𝑏) = (𝑏 +o 1o) ↔ ¬
(1o +o ω) = (ω +o
1o))) |
| 9 | 8 | rspcev 3622 |
. . . . 5
⊢ ((ω
∈ On ∧ ¬ (1o +o ω) = (ω
+o 1o)) → ∃𝑏 ∈ On ¬ (1o +o
𝑏) = (𝑏 +o
1o)) |
| 10 | 4, 9 | mpan 690 |
. . . 4
⊢ (¬
(1o +o ω) = (ω +o 1o)
→ ∃𝑏 ∈ On
¬ (1o +o 𝑏) = (𝑏 +o
1o)) |
| 11 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑎 = 1o → (𝑎 +o 𝑏) = (1o +o
𝑏)) |
| 12 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑎 = 1o → (𝑏 +o 𝑎) = (𝑏 +o
1o)) |
| 13 | 11, 12 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑎 = 1o → ((𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ (1o +o 𝑏) = (𝑏 +o
1o))) |
| 14 | 13 | notbid 318 |
. . . . . 6
⊢ (𝑎 = 1o → (¬
(𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ ¬ (1o +o
𝑏) = (𝑏 +o
1o))) |
| 15 | 14 | rexbidv 3179 |
. . . . 5
⊢ (𝑎 = 1o →
(∃𝑏 ∈ On ¬
(𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ ∃𝑏 ∈ On ¬ (1o +o
𝑏) = (𝑏 +o
1o))) |
| 16 | 15 | rspcev 3622 |
. . . 4
⊢
((1o ∈ On ∧ ∃𝑏 ∈ On ¬ (1o +o
𝑏) = (𝑏 +o 1o)) →
∃𝑎 ∈ On
∃𝑏 ∈ On ¬
(𝑎 +o 𝑏) = (𝑏 +o 𝑎)) |
| 17 | 3, 10, 16 | sylancr 587 |
. . 3
⊢ (¬
(1o +o ω) = (ω +o 1o)
→ ∃𝑎 ∈ On
∃𝑏 ∈ On ¬
(𝑎 +o 𝑏) = (𝑏 +o 𝑎)) |
| 18 | 2, 17 | ax-mp 5 |
. 2
⊢
∃𝑎 ∈ On
∃𝑏 ∈ On ¬
(𝑎 +o 𝑏) = (𝑏 +o 𝑎) |
| 19 | 4, 4 | pm3.2i 470 |
. . . . . . 7
⊢ (ω
∈ On ∧ ω ∈ On) |
| 20 | | peano1 7910 |
. . . . . . 7
⊢ ∅
∈ ω |
| 21 | 19, 20 | pm3.2i 470 |
. . . . . 6
⊢ ((ω
∈ On ∧ ω ∈ On) ∧ ∅ ∈
ω) |
| 22 | | oaord1 8589 |
. . . . . . 7
⊢ ((ω
∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔
ω ∈ (ω +o ω))) |
| 23 | 22 | biimpa 476 |
. . . . . 6
⊢
(((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈
ω) → ω ∈ (ω +o
ω)) |
| 24 | | elneq 9638 |
. . . . . 6
⊢ (ω
∈ (ω +o ω) → ω ≠ (ω
+o ω)) |
| 25 | 21, 23, 24 | mp2b 10 |
. . . . 5
⊢ ω
≠ (ω +o ω) |
| 26 | | 2omomeqom 43316 |
. . . . . 6
⊢
(2o ·o ω) = ω |
| 27 | | df-2o 8507 |
. . . . . . . 8
⊢
2o = suc 1o |
| 28 | 27 | oveq2i 7442 |
. . . . . . 7
⊢ (ω
·o 2o) = (ω ·o suc
1o) |
| 29 | | omsuc 8564 |
. . . . . . . 8
⊢ ((ω
∈ On ∧ 1o ∈ On) → (ω ·o
suc 1o) = ((ω ·o 1o)
+o ω)) |
| 30 | 4, 3, 29 | mp2an 692 |
. . . . . . 7
⊢ (ω
·o suc 1o) = ((ω ·o
1o) +o ω) |
| 31 | | om1 8580 |
. . . . . . . . 9
⊢ (ω
∈ On → (ω ·o 1o) =
ω) |
| 32 | 4, 31 | ax-mp 5 |
. . . . . . . 8
⊢ (ω
·o 1o) = ω |
| 33 | 32 | oveq1i 7441 |
. . . . . . 7
⊢ ((ω
·o 1o) +o ω) = (ω
+o ω) |
| 34 | 28, 30, 33 | 3eqtri 2769 |
. . . . . 6
⊢ (ω
·o 2o) = (ω +o
ω) |
| 35 | 26, 34 | neeq12i 3007 |
. . . . 5
⊢
((2o ·o ω) ≠ (ω
·o 2o) ↔ ω ≠ (ω +o
ω)) |
| 36 | 25, 35 | mpbir 231 |
. . . 4
⊢
(2o ·o ω) ≠ (ω
·o 2o) |
| 37 | 36 | neii 2942 |
. . 3
⊢ ¬
(2o ·o ω) = (ω ·o
2o) |
| 38 | | 2on 8520 |
. . . 4
⊢
2o ∈ On |
| 39 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑏 = ω → (2o
·o 𝑏) =
(2o ·o ω)) |
| 40 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑏 = ω → (𝑏 ·o
2o) = (ω ·o 2o)) |
| 41 | 39, 40 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑏 = ω →
((2o ·o 𝑏) = (𝑏 ·o 2o) ↔
(2o ·o ω) = (ω ·o
2o))) |
| 42 | 41 | notbid 318 |
. . . . . 6
⊢ (𝑏 = ω → (¬
(2o ·o 𝑏) = (𝑏 ·o 2o) ↔
¬ (2o ·o ω) = (ω
·o 2o))) |
| 43 | 42 | rspcev 3622 |
. . . . 5
⊢ ((ω
∈ On ∧ ¬ (2o ·o ω) = (ω
·o 2o)) → ∃𝑏 ∈ On ¬ (2o
·o 𝑏) =
(𝑏 ·o
2o)) |
| 44 | 4, 43 | mpan 690 |
. . . 4
⊢ (¬
(2o ·o ω) = (ω ·o
2o) → ∃𝑏 ∈ On ¬ (2o
·o 𝑏) =
(𝑏 ·o
2o)) |
| 45 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑎 = 2o → (𝑎 ·o 𝑏) = (2o
·o 𝑏)) |
| 46 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑎 = 2o → (𝑏 ·o 𝑎) = (𝑏 ·o
2o)) |
| 47 | 45, 46 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑎 = 2o → ((𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ↔ (2o ·o
𝑏) = (𝑏 ·o
2o))) |
| 48 | 47 | notbid 318 |
. . . . . 6
⊢ (𝑎 = 2o → (¬
(𝑎 ·o
𝑏) = (𝑏 ·o 𝑎) ↔ ¬ (2o
·o 𝑏) =
(𝑏 ·o
2o))) |
| 49 | 48 | rexbidv 3179 |
. . . . 5
⊢ (𝑎 = 2o →
(∃𝑏 ∈ On ¬
(𝑎 ·o
𝑏) = (𝑏 ·o 𝑎) ↔ ∃𝑏 ∈ On ¬ (2o
·o 𝑏) =
(𝑏 ·o
2o))) |
| 50 | 49 | rspcev 3622 |
. . . 4
⊢
((2o ∈ On ∧ ∃𝑏 ∈ On ¬ (2o
·o 𝑏) =
(𝑏 ·o
2o)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎)) |
| 51 | 38, 44, 50 | sylancr 587 |
. . 3
⊢ (¬
(2o ·o ω) = (ω ·o
2o) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎)) |
| 52 | 37, 51 | ax-mp 5 |
. 2
⊢
∃𝑎 ∈ On
∃𝑏 ∈ On ¬
(𝑎 ·o
𝑏) = (𝑏 ·o 𝑎) |
| 53 | | 1onn 8678 |
. . . . . . 7
⊢
1o ∈ ω |
| 54 | 21, 53 | pm3.2i 470 |
. . . . . 6
⊢
(((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈
ω) ∧ 1o ∈ ω) |
| 55 | 4, 31 | mp1i 13 |
. . . . . . 7
⊢
((((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈
ω) ∧ 1o ∈ ω) → (ω
·o 1o) = ω) |
| 56 | | omordi 8604 |
. . . . . . . 8
⊢
(((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈
ω) → (1o ∈ ω → (ω
·o 1o) ∈ (ω ·o
ω))) |
| 57 | 56 | imp 406 |
. . . . . . 7
⊢
((((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈
ω) ∧ 1o ∈ ω) → (ω
·o 1o) ∈ (ω ·o
ω)) |
| 58 | 55, 57 | eqeltrrd 2842 |
. . . . . 6
⊢
((((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈
ω) ∧ 1o ∈ ω) → ω ∈ (ω
·o ω)) |
| 59 | | elneq 9638 |
. . . . . 6
⊢ (ω
∈ (ω ·o ω) → ω ≠ (ω
·o ω)) |
| 60 | 54, 58, 59 | mp2b 10 |
. . . . 5
⊢ ω
≠ (ω ·o ω) |
| 61 | | 2onn 8680 |
. . . . . . 7
⊢
2o ∈ ω |
| 62 | | 1oex 8516 |
. . . . . . . . 9
⊢
1o ∈ V |
| 63 | 62 | prid2 4763 |
. . . . . . . 8
⊢
1o ∈ {∅, 1o} |
| 64 | | df2o3 8514 |
. . . . . . . 8
⊢
2o = {∅, 1o} |
| 65 | 63, 64 | eleqtrri 2840 |
. . . . . . 7
⊢
1o ∈ 2o |
| 66 | | nnoeomeqom 43325 |
. . . . . . 7
⊢
((2o ∈ ω ∧ 1o ∈ 2o)
→ (2o ↑o ω) = ω) |
| 67 | 61, 65, 66 | mp2an 692 |
. . . . . 6
⊢
(2o ↑o ω) = ω |
| 68 | 27 | oveq2i 7442 |
. . . . . . 7
⊢ (ω
↑o 2o) = (ω ↑o suc
1o) |
| 69 | | oesuc 8565 |
. . . . . . . 8
⊢ ((ω
∈ On ∧ 1o ∈ On) → (ω ↑o suc
1o) = ((ω ↑o 1o)
·o ω)) |
| 70 | 4, 3, 69 | mp2an 692 |
. . . . . . 7
⊢ (ω
↑o suc 1o) = ((ω ↑o
1o) ·o ω) |
| 71 | | oe1 8582 |
. . . . . . . . 9
⊢ (ω
∈ On → (ω ↑o 1o) =
ω) |
| 72 | 4, 71 | ax-mp 5 |
. . . . . . . 8
⊢ (ω
↑o 1o) = ω |
| 73 | 72 | oveq1i 7441 |
. . . . . . 7
⊢ ((ω
↑o 1o) ·o ω) = (ω
·o ω) |
| 74 | 68, 70, 73 | 3eqtri 2769 |
. . . . . 6
⊢ (ω
↑o 2o) = (ω ·o
ω) |
| 75 | 67, 74 | neeq12i 3007 |
. . . . 5
⊢
((2o ↑o ω) ≠ (ω
↑o 2o) ↔ ω ≠ (ω
·o ω)) |
| 76 | 60, 75 | mpbir 231 |
. . . 4
⊢
(2o ↑o ω) ≠ (ω
↑o 2o) |
| 77 | 76 | neii 2942 |
. . 3
⊢ ¬
(2o ↑o ω) = (ω ↑o
2o) |
| 78 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑏 = ω → (2o
↑o 𝑏) =
(2o ↑o ω)) |
| 79 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑏 = ω → (𝑏 ↑o
2o) = (ω ↑o 2o)) |
| 80 | 78, 79 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑏 = ω →
((2o ↑o 𝑏) = (𝑏 ↑o 2o) ↔
(2o ↑o ω) = (ω ↑o
2o))) |
| 81 | 80 | notbid 318 |
. . . . . 6
⊢ (𝑏 = ω → (¬
(2o ↑o 𝑏) = (𝑏 ↑o 2o) ↔
¬ (2o ↑o ω) = (ω ↑o
2o))) |
| 82 | 81 | rspcev 3622 |
. . . . 5
⊢ ((ω
∈ On ∧ ¬ (2o ↑o ω) = (ω
↑o 2o)) → ∃𝑏 ∈ On ¬ (2o
↑o 𝑏) =
(𝑏 ↑o
2o)) |
| 83 | 4, 82 | mpan 690 |
. . . 4
⊢ (¬
(2o ↑o ω) = (ω ↑o
2o) → ∃𝑏 ∈ On ¬ (2o
↑o 𝑏) =
(𝑏 ↑o
2o)) |
| 84 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑎 = 2o → (𝑎 ↑o 𝑏) = (2o
↑o 𝑏)) |
| 85 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑎 = 2o → (𝑏 ↑o 𝑎) = (𝑏 ↑o
2o)) |
| 86 | 84, 85 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑎 = 2o → ((𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎) ↔ (2o ↑o
𝑏) = (𝑏 ↑o
2o))) |
| 87 | 86 | notbid 318 |
. . . . . 6
⊢ (𝑎 = 2o → (¬
(𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎) ↔ ¬ (2o
↑o 𝑏) =
(𝑏 ↑o
2o))) |
| 88 | 87 | rexbidv 3179 |
. . . . 5
⊢ (𝑎 = 2o →
(∃𝑏 ∈ On ¬
(𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎) ↔ ∃𝑏 ∈ On ¬ (2o
↑o 𝑏) =
(𝑏 ↑o
2o))) |
| 89 | 88 | rspcev 3622 |
. . . 4
⊢
((2o ∈ On ∧ ∃𝑏 ∈ On ¬ (2o
↑o 𝑏) =
(𝑏 ↑o
2o)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎)) |
| 90 | 38, 83, 89 | sylancr 587 |
. . 3
⊢ (¬
(2o ↑o ω) = (ω ↑o
2o) → ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎)) |
| 91 | 77, 90 | ax-mp 5 |
. 2
⊢
∃𝑎 ∈ On
∃𝑏 ∈ On ¬
(𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎) |
| 92 | 18, 52, 91 | 3pm3.2i 1340 |
1
⊢
(∃𝑎 ∈ On
∃𝑏 ∈ On ¬
(𝑎 +o 𝑏) = (𝑏 +o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎)) |