Step |
Hyp |
Ref |
Expression |
1 |
|
reldom |
⊢ Rel ≼ |
2 |
1
|
brrelex2i |
⊢ ( ω ≼ 𝐴 → 𝐴 ∈ V ) |
3 |
|
djudoml |
⊢ ( ( 𝐴 ∈ V ∧ 𝐴 ∈ V ) → 𝐴 ≼ ( 𝐴 ⊔ 𝐴 ) ) |
4 |
2 2 3
|
syl2anc |
⊢ ( ω ≼ 𝐴 → 𝐴 ≼ ( 𝐴 ⊔ 𝐴 ) ) |
5 |
|
domtr |
⊢ ( ( ω ≼ 𝐴 ∧ 𝐴 ≼ ( 𝐴 ⊔ 𝐴 ) ) → ω ≼ ( 𝐴 ⊔ 𝐴 ) ) |
6 |
4 5
|
mpdan |
⊢ ( ω ≼ 𝐴 → ω ≼ ( 𝐴 ⊔ 𝐴 ) ) |
7 |
1
|
brrelex2i |
⊢ ( ω ≼ ( 𝐴 ⊔ 𝐴 ) → ( 𝐴 ⊔ 𝐴 ) ∈ V ) |
8 |
|
anidm |
⊢ ( ( 𝐴 ∈ V ∧ 𝐴 ∈ V ) ↔ 𝐴 ∈ V ) |
9 |
|
djuexb |
⊢ ( ( 𝐴 ∈ V ∧ 𝐴 ∈ V ) ↔ ( 𝐴 ⊔ 𝐴 ) ∈ V ) |
10 |
8 9
|
bitr3i |
⊢ ( 𝐴 ∈ V ↔ ( 𝐴 ⊔ 𝐴 ) ∈ V ) |
11 |
7 10
|
sylibr |
⊢ ( ω ≼ ( 𝐴 ⊔ 𝐴 ) → 𝐴 ∈ V ) |
12 |
|
domeng |
⊢ ( ( 𝐴 ⊔ 𝐴 ) ∈ V → ( ω ≼ ( 𝐴 ⊔ 𝐴 ) ↔ ∃ 𝑥 ( ω ≈ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ⊔ 𝐴 ) ) ) ) |
13 |
7 12
|
syl |
⊢ ( ω ≼ ( 𝐴 ⊔ 𝐴 ) → ( ω ≼ ( 𝐴 ⊔ 𝐴 ) ↔ ∃ 𝑥 ( ω ≈ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ⊔ 𝐴 ) ) ) ) |
14 |
13
|
ibi |
⊢ ( ω ≼ ( 𝐴 ⊔ 𝐴 ) → ∃ 𝑥 ( ω ≈ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ⊔ 𝐴 ) ) ) |
15 |
|
indi |
⊢ ( 𝑥 ∩ ( ( { ∅ } × 𝐴 ) ∪ ( { 1o } × 𝐴 ) ) ) = ( ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ∪ ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ) |
16 |
|
simpr |
⊢ ( ( ω ≈ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ⊔ 𝐴 ) ) → 𝑥 ⊆ ( 𝐴 ⊔ 𝐴 ) ) |
17 |
|
df-dju |
⊢ ( 𝐴 ⊔ 𝐴 ) = ( ( { ∅ } × 𝐴 ) ∪ ( { 1o } × 𝐴 ) ) |
18 |
16 17
|
sseqtrdi |
⊢ ( ( ω ≈ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ⊔ 𝐴 ) ) → 𝑥 ⊆ ( ( { ∅ } × 𝐴 ) ∪ ( { 1o } × 𝐴 ) ) ) |
19 |
|
df-ss |
⊢ ( 𝑥 ⊆ ( ( { ∅ } × 𝐴 ) ∪ ( { 1o } × 𝐴 ) ) ↔ ( 𝑥 ∩ ( ( { ∅ } × 𝐴 ) ∪ ( { 1o } × 𝐴 ) ) ) = 𝑥 ) |
20 |
18 19
|
sylib |
⊢ ( ( ω ≈ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ⊔ 𝐴 ) ) → ( 𝑥 ∩ ( ( { ∅ } × 𝐴 ) ∪ ( { 1o } × 𝐴 ) ) ) = 𝑥 ) |
21 |
15 20
|
eqtr3id |
⊢ ( ( ω ≈ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ⊔ 𝐴 ) ) → ( ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ∪ ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ) = 𝑥 ) |
22 |
|
ensym |
⊢ ( ω ≈ 𝑥 → 𝑥 ≈ ω ) |
23 |
22
|
adantr |
⊢ ( ( ω ≈ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ⊔ 𝐴 ) ) → 𝑥 ≈ ω ) |
24 |
21 23
|
eqbrtrd |
⊢ ( ( ω ≈ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ⊔ 𝐴 ) ) → ( ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ∪ ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ) ≈ ω ) |
25 |
|
cdainflem |
⊢ ( ( ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ∪ ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ) ≈ ω → ( ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ≈ ω ∨ ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ≈ ω ) ) |
26 |
|
snex |
⊢ { ∅ } ∈ V |
27 |
|
xpexg |
⊢ ( ( { ∅ } ∈ V ∧ 𝐴 ∈ V ) → ( { ∅ } × 𝐴 ) ∈ V ) |
28 |
26 27
|
mpan |
⊢ ( 𝐴 ∈ V → ( { ∅ } × 𝐴 ) ∈ V ) |
29 |
|
inss2 |
⊢ ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ⊆ ( { ∅ } × 𝐴 ) |
30 |
|
ssdomg |
⊢ ( ( { ∅ } × 𝐴 ) ∈ V → ( ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ⊆ ( { ∅ } × 𝐴 ) → ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ≼ ( { ∅ } × 𝐴 ) ) ) |
31 |
28 29 30
|
mpisyl |
⊢ ( 𝐴 ∈ V → ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ≼ ( { ∅ } × 𝐴 ) ) |
32 |
|
0ex |
⊢ ∅ ∈ V |
33 |
|
xpsnen2g |
⊢ ( ( ∅ ∈ V ∧ 𝐴 ∈ V ) → ( { ∅ } × 𝐴 ) ≈ 𝐴 ) |
34 |
32 33
|
mpan |
⊢ ( 𝐴 ∈ V → ( { ∅ } × 𝐴 ) ≈ 𝐴 ) |
35 |
|
domentr |
⊢ ( ( ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ≼ ( { ∅ } × 𝐴 ) ∧ ( { ∅ } × 𝐴 ) ≈ 𝐴 ) → ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ≼ 𝐴 ) |
36 |
31 34 35
|
syl2anc |
⊢ ( 𝐴 ∈ V → ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ≼ 𝐴 ) |
37 |
|
domen1 |
⊢ ( ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ≈ ω → ( ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ≼ 𝐴 ↔ ω ≼ 𝐴 ) ) |
38 |
36 37
|
syl5ibcom |
⊢ ( 𝐴 ∈ V → ( ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ≈ ω → ω ≼ 𝐴 ) ) |
39 |
|
snex |
⊢ { 1o } ∈ V |
40 |
|
xpexg |
⊢ ( ( { 1o } ∈ V ∧ 𝐴 ∈ V ) → ( { 1o } × 𝐴 ) ∈ V ) |
41 |
39 40
|
mpan |
⊢ ( 𝐴 ∈ V → ( { 1o } × 𝐴 ) ∈ V ) |
42 |
|
inss2 |
⊢ ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ⊆ ( { 1o } × 𝐴 ) |
43 |
|
ssdomg |
⊢ ( ( { 1o } × 𝐴 ) ∈ V → ( ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ⊆ ( { 1o } × 𝐴 ) → ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ≼ ( { 1o } × 𝐴 ) ) ) |
44 |
41 42 43
|
mpisyl |
⊢ ( 𝐴 ∈ V → ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ≼ ( { 1o } × 𝐴 ) ) |
45 |
|
1on |
⊢ 1o ∈ On |
46 |
|
xpsnen2g |
⊢ ( ( 1o ∈ On ∧ 𝐴 ∈ V ) → ( { 1o } × 𝐴 ) ≈ 𝐴 ) |
47 |
45 46
|
mpan |
⊢ ( 𝐴 ∈ V → ( { 1o } × 𝐴 ) ≈ 𝐴 ) |
48 |
|
domentr |
⊢ ( ( ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ≼ ( { 1o } × 𝐴 ) ∧ ( { 1o } × 𝐴 ) ≈ 𝐴 ) → ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ≼ 𝐴 ) |
49 |
44 47 48
|
syl2anc |
⊢ ( 𝐴 ∈ V → ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ≼ 𝐴 ) |
50 |
|
domen1 |
⊢ ( ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ≈ ω → ( ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ≼ 𝐴 ↔ ω ≼ 𝐴 ) ) |
51 |
49 50
|
syl5ibcom |
⊢ ( 𝐴 ∈ V → ( ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ≈ ω → ω ≼ 𝐴 ) ) |
52 |
38 51
|
jaod |
⊢ ( 𝐴 ∈ V → ( ( ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ≈ ω ∨ ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ≈ ω ) → ω ≼ 𝐴 ) ) |
53 |
25 52
|
syl5 |
⊢ ( 𝐴 ∈ V → ( ( ( 𝑥 ∩ ( { ∅ } × 𝐴 ) ) ∪ ( 𝑥 ∩ ( { 1o } × 𝐴 ) ) ) ≈ ω → ω ≼ 𝐴 ) ) |
54 |
24 53
|
syl5 |
⊢ ( 𝐴 ∈ V → ( ( ω ≈ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ⊔ 𝐴 ) ) → ω ≼ 𝐴 ) ) |
55 |
54
|
exlimdv |
⊢ ( 𝐴 ∈ V → ( ∃ 𝑥 ( ω ≈ 𝑥 ∧ 𝑥 ⊆ ( 𝐴 ⊔ 𝐴 ) ) → ω ≼ 𝐴 ) ) |
56 |
11 14 55
|
sylc |
⊢ ( ω ≼ ( 𝐴 ⊔ 𝐴 ) → ω ≼ 𝐴 ) |
57 |
6 56
|
impbii |
⊢ ( ω ≼ 𝐴 ↔ ω ≼ ( 𝐴 ⊔ 𝐴 ) ) |