Proof of Theorem oaomoencom
Step | Hyp | Ref
| Expression |
1 | | oancom 9630 |
. . . 4
⊢
(1o +o ω) ≠ (ω +o
1o) |
2 | 1 | neii 2942 |
. . 3
⊢ ¬
(1o +o ω) = (ω +o
1o) |
3 | | 1on 8462 |
. . . 4
⊢
1o ∈ On |
4 | | omelon 9625 |
. . . . 5
⊢ ω
∈ On |
5 | | oveq2 7402 |
. . . . . . . 8
⊢ (𝑏 = ω → (1o
+o 𝑏) =
(1o +o ω)) |
6 | | oveq1 7401 |
. . . . . . . 8
⊢ (𝑏 = ω → (𝑏 +o 1o) =
(ω +o 1o)) |
7 | 5, 6 | eqeq12d 2748 |
. . . . . . 7
⊢ (𝑏 = ω →
((1o +o 𝑏) = (𝑏 +o 1o) ↔
(1o +o ω) = (ω +o
1o))) |
8 | 7 | notbid 317 |
. . . . . 6
⊢ (𝑏 = ω → (¬
(1o +o 𝑏) = (𝑏 +o 1o) ↔ ¬
(1o +o ω) = (ω +o
1o))) |
9 | 8 | rspcev 3610 |
. . . . 5
⊢ ((ω
∈ On ∧ ¬ (1o +o ω) = (ω
+o 1o)) → ∃𝑏 ∈ On ¬ (1o +o
𝑏) = (𝑏 +o
1o)) |
10 | 4, 9 | mpan 688 |
. . . 4
⊢ (¬
(1o +o ω) = (ω +o 1o)
→ ∃𝑏 ∈ On
¬ (1o +o 𝑏) = (𝑏 +o
1o)) |
11 | | oveq1 7401 |
. . . . . . . 8
⊢ (𝑎 = 1o → (𝑎 +o 𝑏) = (1o +o
𝑏)) |
12 | | oveq2 7402 |
. . . . . . . 8
⊢ (𝑎 = 1o → (𝑏 +o 𝑎) = (𝑏 +o
1o)) |
13 | 11, 12 | eqeq12d 2748 |
. . . . . . 7
⊢ (𝑎 = 1o → ((𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ (1o +o 𝑏) = (𝑏 +o
1o))) |
14 | 13 | notbid 317 |
. . . . . 6
⊢ (𝑎 = 1o → (¬
(𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ ¬ (1o +o
𝑏) = (𝑏 +o
1o))) |
15 | 14 | rexbidv 3178 |
. . . . 5
⊢ (𝑎 = 1o →
(∃𝑏 ∈ On ¬
(𝑎 +o 𝑏) = (𝑏 +o 𝑎) ↔ ∃𝑏 ∈ On ¬ (1o +o
𝑏) = (𝑏 +o
1o))) |
16 | 15 | rspcev 3610 |
. . . 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 471 |
. . . . . . 7
⊢ (ω
∈ On ∧ ω ∈ On) |
20 | | peano1 7863 |
. . . . . . 7
⊢ ∅
∈ ω |
21 | 19, 20 | pm3.2i 471 |
. . . . . 6
⊢ ((ω
∈ On ∧ ω ∈ On) ∧ ∅ ∈
ω) |
22 | | oaord1 8536 |
. . . . . . 7
⊢ ((ω
∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔
ω ∈ (ω +o ω))) |
23 | 22 | biimpa 477 |
. . . . . 6
⊢
(((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈
ω) → ω ∈ (ω +o
ω)) |
24 | | elneq 9577 |
. . . . . 6
⊢ (ω
∈ (ω +o ω) → ω ≠ (ω
+o ω)) |
25 | 21, 23, 24 | mp2b 10 |
. . . . 5
⊢ ω
≠ (ω +o ω) |
26 | | 2omomeqom 41888 |
. . . . . 6
⊢
(2o ·o ω) = ω |
27 | | df-2o 8451 |
. . . . . . . 8
⊢
2o = suc 1o |
28 | 27 | oveq2i 7405 |
. . . . . . 7
⊢ (ω
·o 2o) = (ω ·o suc
1o) |
29 | | omsuc 8510 |
. . . . . . . 8
⊢ ((ω
∈ On ∧ 1o ∈ On) → (ω ·o
suc 1o) = ((ω ·o 1o)
+o ω)) |
30 | 4, 3, 29 | mp2an 690 |
. . . . . . 7
⊢ (ω
·o suc 1o) = ((ω ·o
1o) +o ω) |
31 | | om1 8527 |
. . . . . . . . 9
⊢ (ω
∈ On → (ω ·o 1o) =
ω) |
32 | 4, 31 | ax-mp 5 |
. . . . . . . 8
⊢ (ω
·o 1o) = ω |
33 | 32 | oveq1i 7404 |
. . . . . . 7
⊢ ((ω
·o 1o) +o ω) = (ω
+o ω) |
34 | 28, 30, 33 | 3eqtri 2764 |
. . . . . 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 230 |
. . . 4
⊢
(2o ·o ω) ≠ (ω
·o 2o) |
37 | 36 | neii 2942 |
. . 3
⊢ ¬
(2o ·o ω) = (ω ·o
2o) |
38 | | 2on 8464 |
. . . 4
⊢
2o ∈ On |
39 | | oveq2 7402 |
. . . . . . . 8
⊢ (𝑏 = ω → (2o
·o 𝑏) =
(2o ·o ω)) |
40 | | oveq1 7401 |
. . . . . . . 8
⊢ (𝑏 = ω → (𝑏 ·o
2o) = (ω ·o 2o)) |
41 | 39, 40 | eqeq12d 2748 |
. . . . . . 7
⊢ (𝑏 = ω →
((2o ·o 𝑏) = (𝑏 ·o 2o) ↔
(2o ·o ω) = (ω ·o
2o))) |
42 | 41 | notbid 317 |
. . . . . 6
⊢ (𝑏 = ω → (¬
(2o ·o 𝑏) = (𝑏 ·o 2o) ↔
¬ (2o ·o ω) = (ω
·o 2o))) |
43 | 42 | rspcev 3610 |
. . . . 5
⊢ ((ω
∈ On ∧ ¬ (2o ·o ω) = (ω
·o 2o)) → ∃𝑏 ∈ On ¬ (2o
·o 𝑏) =
(𝑏 ·o
2o)) |
44 | 4, 43 | mpan 688 |
. . . 4
⊢ (¬
(2o ·o ω) = (ω ·o
2o) → ∃𝑏 ∈ On ¬ (2o
·o 𝑏) =
(𝑏 ·o
2o)) |
45 | | oveq1 7401 |
. . . . . . . 8
⊢ (𝑎 = 2o → (𝑎 ·o 𝑏) = (2o
·o 𝑏)) |
46 | | oveq2 7402 |
. . . . . . . 8
⊢ (𝑎 = 2o → (𝑏 ·o 𝑎) = (𝑏 ·o
2o)) |
47 | 45, 46 | eqeq12d 2748 |
. . . . . . 7
⊢ (𝑎 = 2o → ((𝑎 ·o 𝑏) = (𝑏 ·o 𝑎) ↔ (2o ·o
𝑏) = (𝑏 ·o
2o))) |
48 | 47 | notbid 317 |
. . . . . 6
⊢ (𝑎 = 2o → (¬
(𝑎 ·o
𝑏) = (𝑏 ·o 𝑎) ↔ ¬ (2o
·o 𝑏) =
(𝑏 ·o
2o))) |
49 | 48 | rexbidv 3178 |
. . . . 5
⊢ (𝑎 = 2o →
(∃𝑏 ∈ On ¬
(𝑎 ·o
𝑏) = (𝑏 ·o 𝑎) ↔ ∃𝑏 ∈ On ¬ (2o
·o 𝑏) =
(𝑏 ·o
2o))) |
50 | 49 | rspcev 3610 |
. . . 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 8624 |
. . . . . . 7
⊢
1o ∈ ω |
54 | 21, 53 | pm3.2i 471 |
. . . . . 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 8551 |
. . . . . . . 8
⊢
(((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈
ω) → (1o ∈ ω → (ω
·o 1o) ∈ (ω ·o
ω))) |
57 | 56 | imp 407 |
. . . . . . 7
⊢
((((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈
ω) ∧ 1o ∈ ω) → (ω
·o 1o) ∈ (ω ·o
ω)) |
58 | 55, 57 | eqeltrrd 2834 |
. . . . . 6
⊢
((((ω ∈ On ∧ ω ∈ On) ∧ ∅ ∈
ω) ∧ 1o ∈ ω) → ω ∈ (ω
·o ω)) |
59 | | elneq 9577 |
. . . . . 6
⊢ (ω
∈ (ω ·o ω) → ω ≠ (ω
·o ω)) |
60 | 54, 58, 59 | mp2b 10 |
. . . . 5
⊢ ω
≠ (ω ·o ω) |
61 | | 2onn 8626 |
. . . . . . 7
⊢
2o ∈ ω |
62 | | 1oex 8460 |
. . . . . . . . 9
⊢
1o ∈ V |
63 | 62 | prid2 4761 |
. . . . . . . 8
⊢
1o ∈ {∅, 1o} |
64 | | df2o3 8458 |
. . . . . . . 8
⊢
2o = {∅, 1o} |
65 | 63, 64 | eleqtrri 2832 |
. . . . . . 7
⊢
1o ∈ 2o |
66 | | nnoeomeqom 41897 |
. . . . . . 7
⊢
((2o ∈ ω ∧ 1o ∈ 2o)
→ (2o ↑o ω) = ω) |
67 | 61, 65, 66 | mp2an 690 |
. . . . . 6
⊢
(2o ↑o ω) = ω |
68 | 27 | oveq2i 7405 |
. . . . . . 7
⊢ (ω
↑o 2o) = (ω ↑o suc
1o) |
69 | | oesuc 8511 |
. . . . . . . 8
⊢ ((ω
∈ On ∧ 1o ∈ On) → (ω ↑o suc
1o) = ((ω ↑o 1o)
·o ω)) |
70 | 4, 3, 69 | mp2an 690 |
. . . . . . 7
⊢ (ω
↑o suc 1o) = ((ω ↑o
1o) ·o ω) |
71 | | oe1 8529 |
. . . . . . . . 9
⊢ (ω
∈ On → (ω ↑o 1o) =
ω) |
72 | 4, 71 | ax-mp 5 |
. . . . . . . 8
⊢ (ω
↑o 1o) = ω |
73 | 72 | oveq1i 7404 |
. . . . . . 7
⊢ ((ω
↑o 1o) ·o ω) = (ω
·o ω) |
74 | 68, 70, 73 | 3eqtri 2764 |
. . . . . 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 230 |
. . . 4
⊢
(2o ↑o ω) ≠ (ω
↑o 2o) |
77 | 76 | neii 2942 |
. . 3
⊢ ¬
(2o ↑o ω) = (ω ↑o
2o) |
78 | | oveq2 7402 |
. . . . . . . 8
⊢ (𝑏 = ω → (2o
↑o 𝑏) =
(2o ↑o ω)) |
79 | | oveq1 7401 |
. . . . . . . 8
⊢ (𝑏 = ω → (𝑏 ↑o
2o) = (ω ↑o 2o)) |
80 | 78, 79 | eqeq12d 2748 |
. . . . . . 7
⊢ (𝑏 = ω →
((2o ↑o 𝑏) = (𝑏 ↑o 2o) ↔
(2o ↑o ω) = (ω ↑o
2o))) |
81 | 80 | notbid 317 |
. . . . . 6
⊢ (𝑏 = ω → (¬
(2o ↑o 𝑏) = (𝑏 ↑o 2o) ↔
¬ (2o ↑o ω) = (ω ↑o
2o))) |
82 | 81 | rspcev 3610 |
. . . . 5
⊢ ((ω
∈ On ∧ ¬ (2o ↑o ω) = (ω
↑o 2o)) → ∃𝑏 ∈ On ¬ (2o
↑o 𝑏) =
(𝑏 ↑o
2o)) |
83 | 4, 82 | mpan 688 |
. . . 4
⊢ (¬
(2o ↑o ω) = (ω ↑o
2o) → ∃𝑏 ∈ On ¬ (2o
↑o 𝑏) =
(𝑏 ↑o
2o)) |
84 | | oveq1 7401 |
. . . . . . . 8
⊢ (𝑎 = 2o → (𝑎 ↑o 𝑏) = (2o
↑o 𝑏)) |
85 | | oveq2 7402 |
. . . . . . . 8
⊢ (𝑎 = 2o → (𝑏 ↑o 𝑎) = (𝑏 ↑o
2o)) |
86 | 84, 85 | eqeq12d 2748 |
. . . . . . 7
⊢ (𝑎 = 2o → ((𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎) ↔ (2o ↑o
𝑏) = (𝑏 ↑o
2o))) |
87 | 86 | notbid 317 |
. . . . . 6
⊢ (𝑎 = 2o → (¬
(𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎) ↔ ¬ (2o
↑o 𝑏) =
(𝑏 ↑o
2o))) |
88 | 87 | rexbidv 3178 |
. . . . 5
⊢ (𝑎 = 2o →
(∃𝑏 ∈ On ¬
(𝑎 ↑o 𝑏) = (𝑏 ↑o 𝑎) ↔ ∃𝑏 ∈ On ¬ (2o
↑o 𝑏) =
(𝑏 ↑o
2o))) |
89 | 88 | rspcev 3610 |
. . . 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 1339 |
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)) |