Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( ω × { 𝑋 } ) = ( ω × { 𝑋 } ) |
2 |
|
omex |
⊢ ω ∈ V |
3 |
|
snex |
⊢ { 𝑋 } ∈ V |
4 |
2 3
|
xpex |
⊢ ( ω × { 𝑋 } ) ∈ V |
5 |
|
eqeq1 |
⊢ ( 𝑎 = ( ω × { 𝑋 } ) → ( 𝑎 = ( ω × { 𝑋 } ) ↔ ( ω × { 𝑋 } ) = ( ω × { 𝑋 } ) ) ) |
6 |
4 5
|
spcev |
⊢ ( ( ω × { 𝑋 } ) = ( ω × { 𝑋 } ) → ∃ 𝑎 𝑎 = ( ω × { 𝑋 } ) ) |
7 |
1 6
|
mp1i |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ∃ 𝑎 𝑎 = ( ω × { 𝑋 } ) ) |
8 |
3 2
|
pm3.2i |
⊢ ( { 𝑋 } ∈ V ∧ ω ∈ V ) |
9 |
|
elmapg |
⊢ ( ( { 𝑋 } ∈ V ∧ ω ∈ V ) → ( 𝑎 ∈ ( { 𝑋 } ↑m ω ) ↔ 𝑎 : ω ⟶ { 𝑋 } ) ) |
10 |
8 9
|
mp1i |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ( 𝑎 ∈ ( { 𝑋 } ↑m ω ) ↔ 𝑎 : ω ⟶ { 𝑋 } ) ) |
11 |
|
fconst2g |
⊢ ( 𝑋 ∈ 𝑉 → ( 𝑎 : ω ⟶ { 𝑋 } ↔ 𝑎 = ( ω × { 𝑋 } ) ) ) |
12 |
11
|
3ad2ant3 |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ( 𝑎 : ω ⟶ { 𝑋 } ↔ 𝑎 = ( ω × { 𝑋 } ) ) ) |
13 |
10 12
|
bitrd |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ( 𝑎 ∈ ( { 𝑋 } ↑m ω ) ↔ 𝑎 = ( ω × { 𝑋 } ) ) ) |
14 |
13
|
exbidv |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ( ∃ 𝑎 𝑎 ∈ ( { 𝑋 } ↑m ω ) ↔ ∃ 𝑎 𝑎 = ( ω × { 𝑋 } ) ) ) |
15 |
7 14
|
mpbird |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ∃ 𝑎 𝑎 ∈ ( { 𝑋 } ↑m ω ) ) |
16 |
|
neq0 |
⊢ ( ¬ ( { 𝑋 } ↑m ω ) = ∅ ↔ ∃ 𝑎 𝑎 ∈ ( { 𝑋 } ↑m ω ) ) |
17 |
15 16
|
sylibr |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ¬ ( { 𝑋 } ↑m ω ) = ∅ ) |
18 |
|
eqcom |
⊢ ( ( { 𝑋 } ↑m ω ) = ∅ ↔ ∅ = ( { 𝑋 } ↑m ω ) ) |
19 |
17 18
|
sylnib |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ¬ ∅ = ( { 𝑋 } ↑m ω ) ) |
20 |
|
ovex |
⊢ ( 𝐼 ∈𝑔 𝐽 ) ∈ V |
21 |
3 20
|
pm3.2i |
⊢ ( { 𝑋 } ∈ V ∧ ( 𝐼 ∈𝑔 𝐽 ) ∈ V ) |
22 |
|
prv |
⊢ ( ( { 𝑋 } ∈ V ∧ ( 𝐼 ∈𝑔 𝐽 ) ∈ V ) → ( { 𝑋 } ⊧ ( 𝐼 ∈𝑔 𝐽 ) ↔ ( { 𝑋 } Sat∈ ( 𝐼 ∈𝑔 𝐽 ) ) = ( { 𝑋 } ↑m ω ) ) ) |
23 |
21 22
|
mp1i |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ( { 𝑋 } ⊧ ( 𝐼 ∈𝑔 𝐽 ) ↔ ( { 𝑋 } Sat∈ ( 𝐼 ∈𝑔 𝐽 ) ) = ( { 𝑋 } ↑m ω ) ) ) |
24 |
|
goel |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → ( 𝐼 ∈𝑔 𝐽 ) = 〈 ∅ , 〈 𝐼 , 𝐽 〉 〉 ) |
25 |
|
0ex |
⊢ ∅ ∈ V |
26 |
25
|
snid |
⊢ ∅ ∈ { ∅ } |
27 |
26
|
a1i |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → ∅ ∈ { ∅ } ) |
28 |
|
opelxpi |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → 〈 𝐼 , 𝐽 〉 ∈ ( ω × ω ) ) |
29 |
27 28
|
opelxpd |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → 〈 ∅ , 〈 𝐼 , 𝐽 〉 〉 ∈ ( { ∅ } × ( ω × ω ) ) ) |
30 |
24 29
|
eqeltrd |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → ( 𝐼 ∈𝑔 𝐽 ) ∈ ( { ∅ } × ( ω × ω ) ) ) |
31 |
|
fmla0xp |
⊢ ( Fmla ‘ ∅ ) = ( { ∅ } × ( ω × ω ) ) |
32 |
30 31
|
eleqtrrdi |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → ( 𝐼 ∈𝑔 𝐽 ) ∈ ( Fmla ‘ ∅ ) ) |
33 |
32
|
3adant3 |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ( 𝐼 ∈𝑔 𝐽 ) ∈ ( Fmla ‘ ∅ ) ) |
34 |
|
satefvfmla0 |
⊢ ( ( { 𝑋 } ∈ V ∧ ( 𝐼 ∈𝑔 𝐽 ) ∈ ( Fmla ‘ ∅ ) ) → ( { 𝑋 } Sat∈ ( 𝐼 ∈𝑔 𝐽 ) ) = { 𝑎 ∈ ( { 𝑋 } ↑m ω ) ∣ ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) ) } ) |
35 |
3 33 34
|
sylancr |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ( { 𝑋 } Sat∈ ( 𝐼 ∈𝑔 𝐽 ) ) = { 𝑎 ∈ ( { 𝑋 } ↑m ω ) ∣ ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) ) } ) |
36 |
24
|
fveq2d |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) = ( 2nd ‘ 〈 ∅ , 〈 𝐼 , 𝐽 〉 〉 ) ) |
37 |
|
opex |
⊢ 〈 𝐼 , 𝐽 〉 ∈ V |
38 |
25 37
|
op2nd |
⊢ ( 2nd ‘ 〈 ∅ , 〈 𝐼 , 𝐽 〉 〉 ) = 〈 𝐼 , 𝐽 〉 |
39 |
36 38
|
eqtrdi |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) = 〈 𝐼 , 𝐽 〉 ) |
40 |
39
|
fveq2d |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → ( 1st ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) = ( 1st ‘ 〈 𝐼 , 𝐽 〉 ) ) |
41 |
|
op1stg |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → ( 1st ‘ 〈 𝐼 , 𝐽 〉 ) = 𝐼 ) |
42 |
40 41
|
eqtrd |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → ( 1st ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) = 𝐼 ) |
43 |
42
|
fveq2d |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) ) = ( 𝑎 ‘ 𝐼 ) ) |
44 |
39
|
fveq2d |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → ( 2nd ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) = ( 2nd ‘ 〈 𝐼 , 𝐽 〉 ) ) |
45 |
|
op2ndg |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → ( 2nd ‘ 〈 𝐼 , 𝐽 〉 ) = 𝐽 ) |
46 |
44 45
|
eqtrd |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → ( 2nd ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) = 𝐽 ) |
47 |
46
|
fveq2d |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) ) = ( 𝑎 ‘ 𝐽 ) ) |
48 |
43 47
|
eleq12d |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → ( ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) ) ↔ ( 𝑎 ‘ 𝐼 ) ∈ ( 𝑎 ‘ 𝐽 ) ) ) |
49 |
48
|
rabbidv |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ) → { 𝑎 ∈ ( { 𝑋 } ↑m ω ) ∣ ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) ) } = { 𝑎 ∈ ( { 𝑋 } ↑m ω ) ∣ ( 𝑎 ‘ 𝐼 ) ∈ ( 𝑎 ‘ 𝐽 ) } ) |
50 |
49
|
3adant3 |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → { 𝑎 ∈ ( { 𝑋 } ↑m ω ) ∣ ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) ) } = { 𝑎 ∈ ( { 𝑋 } ↑m ω ) ∣ ( 𝑎 ‘ 𝐼 ) ∈ ( 𝑎 ‘ 𝐽 ) } ) |
51 |
|
elmapi |
⊢ ( 𝑎 ∈ ( { 𝑋 } ↑m ω ) → 𝑎 : ω ⟶ { 𝑋 } ) |
52 |
|
elirr |
⊢ ¬ 𝑋 ∈ 𝑋 |
53 |
|
fvconst |
⊢ ( ( 𝑎 : ω ⟶ { 𝑋 } ∧ 𝐼 ∈ ω ) → ( 𝑎 ‘ 𝐼 ) = 𝑋 ) |
54 |
53
|
3ad2antr1 |
⊢ ( ( 𝑎 : ω ⟶ { 𝑋 } ∧ ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) ) → ( 𝑎 ‘ 𝐼 ) = 𝑋 ) |
55 |
|
fvconst |
⊢ ( ( 𝑎 : ω ⟶ { 𝑋 } ∧ 𝐽 ∈ ω ) → ( 𝑎 ‘ 𝐽 ) = 𝑋 ) |
56 |
55
|
3ad2antr2 |
⊢ ( ( 𝑎 : ω ⟶ { 𝑋 } ∧ ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) ) → ( 𝑎 ‘ 𝐽 ) = 𝑋 ) |
57 |
54 56
|
eleq12d |
⊢ ( ( 𝑎 : ω ⟶ { 𝑋 } ∧ ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑎 ‘ 𝐼 ) ∈ ( 𝑎 ‘ 𝐽 ) ↔ 𝑋 ∈ 𝑋 ) ) |
58 |
52 57
|
mtbiri |
⊢ ( ( 𝑎 : ω ⟶ { 𝑋 } ∧ ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) ) → ¬ ( 𝑎 ‘ 𝐼 ) ∈ ( 𝑎 ‘ 𝐽 ) ) |
59 |
58
|
ex |
⊢ ( 𝑎 : ω ⟶ { 𝑋 } → ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ¬ ( 𝑎 ‘ 𝐼 ) ∈ ( 𝑎 ‘ 𝐽 ) ) ) |
60 |
51 59
|
syl |
⊢ ( 𝑎 ∈ ( { 𝑋 } ↑m ω ) → ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ¬ ( 𝑎 ‘ 𝐼 ) ∈ ( 𝑎 ‘ 𝐽 ) ) ) |
61 |
60
|
impcom |
⊢ ( ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) ∧ 𝑎 ∈ ( { 𝑋 } ↑m ω ) ) → ¬ ( 𝑎 ‘ 𝐼 ) ∈ ( 𝑎 ‘ 𝐽 ) ) |
62 |
61
|
ralrimiva |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ∀ 𝑎 ∈ ( { 𝑋 } ↑m ω ) ¬ ( 𝑎 ‘ 𝐼 ) ∈ ( 𝑎 ‘ 𝐽 ) ) |
63 |
|
rabeq0 |
⊢ ( { 𝑎 ∈ ( { 𝑋 } ↑m ω ) ∣ ( 𝑎 ‘ 𝐼 ) ∈ ( 𝑎 ‘ 𝐽 ) } = ∅ ↔ ∀ 𝑎 ∈ ( { 𝑋 } ↑m ω ) ¬ ( 𝑎 ‘ 𝐼 ) ∈ ( 𝑎 ‘ 𝐽 ) ) |
64 |
62 63
|
sylibr |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → { 𝑎 ∈ ( { 𝑋 } ↑m ω ) ∣ ( 𝑎 ‘ 𝐼 ) ∈ ( 𝑎 ‘ 𝐽 ) } = ∅ ) |
65 |
50 64
|
eqtrd |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → { 𝑎 ∈ ( { 𝑋 } ↑m ω ) ∣ ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐼 ∈𝑔 𝐽 ) ) ) ) } = ∅ ) |
66 |
35 65
|
eqtrd |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ( { 𝑋 } Sat∈ ( 𝐼 ∈𝑔 𝐽 ) ) = ∅ ) |
67 |
66
|
eqeq1d |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ( ( { 𝑋 } Sat∈ ( 𝐼 ∈𝑔 𝐽 ) ) = ( { 𝑋 } ↑m ω ) ↔ ∅ = ( { 𝑋 } ↑m ω ) ) ) |
68 |
23 67
|
bitrd |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ( { 𝑋 } ⊧ ( 𝐼 ∈𝑔 𝐽 ) ↔ ∅ = ( { 𝑋 } ↑m ω ) ) ) |
69 |
19 68
|
mtbird |
⊢ ( ( 𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉 ) → ¬ { 𝑋 } ⊧ ( 𝐼 ∈𝑔 𝐽 ) ) |