Step |
Hyp |
Ref |
Expression |
1 |
|
breq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ≼ 𝐵 ↔ 𝑦 ≼ 𝐵 ) ) |
2 |
|
eleq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝑈 ↔ 𝑦 ∈ 𝑈 ) ) |
3 |
1 2
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈 ) ↔ ( 𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈 ) ) ) |
4 |
3
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( 𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈 ) ) ↔ ( ( 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( 𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈 ) ) ) ) |
5 |
|
breq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ≼ 𝐵 ↔ 𝐴 ≼ 𝐵 ) ) |
6 |
|
eleq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈 ) ) |
7 |
5 6
|
imbi12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈 ) ↔ ( 𝐴 ≼ 𝐵 → 𝐴 ∈ 𝑈 ) ) ) |
8 |
7
|
imbi2d |
⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( 𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈 ) ) ↔ ( ( 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( 𝐴 ≼ 𝐵 → 𝐴 ∈ 𝑈 ) ) ) ) |
9 |
|
r19.21v |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( ( 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( 𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈 ) ) ↔ ( ( 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ∀ 𝑦 ∈ 𝑥 ( 𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈 ) ) ) |
10 |
|
simpl1 |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) → 𝑥 ∈ On ) |
11 |
|
vex |
⊢ 𝑥 ∈ V |
12 |
|
onelss |
⊢ ( 𝑥 ∈ On → ( 𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥 ) ) |
13 |
12
|
imp |
⊢ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ⊆ 𝑥 ) |
14 |
|
ssdomg |
⊢ ( 𝑥 ∈ V → ( 𝑦 ⊆ 𝑥 → 𝑦 ≼ 𝑥 ) ) |
15 |
11 13 14
|
mpsyl |
⊢ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ≼ 𝑥 ) |
16 |
10 15
|
sylan |
⊢ ( ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ≼ 𝑥 ) |
17 |
|
simplr |
⊢ ( ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) ∧ 𝑦 ∈ 𝑥 ) → 𝑥 ≼ 𝐵 ) |
18 |
|
domtr |
⊢ ( ( 𝑦 ≼ 𝑥 ∧ 𝑥 ≼ 𝐵 ) → 𝑦 ≼ 𝐵 ) |
19 |
16 17 18
|
syl2anc |
⊢ ( ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ≼ 𝐵 ) |
20 |
|
pm2.27 |
⊢ ( 𝑦 ≼ 𝐵 → ( ( 𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈 ) → 𝑦 ∈ 𝑈 ) ) |
21 |
19 20
|
syl |
⊢ ( ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) ∧ 𝑦 ∈ 𝑥 ) → ( ( 𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈 ) → 𝑦 ∈ 𝑈 ) ) |
22 |
21
|
ralimdva |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈 ) → ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝑈 ) ) |
23 |
|
dfss3 |
⊢ ( 𝑥 ⊆ 𝑈 ↔ ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝑈 ) |
24 |
|
domeng |
⊢ ( 𝐵 ∈ 𝑈 → ( 𝑥 ≼ 𝐵 ↔ ∃ 𝑦 ( 𝑥 ≈ 𝑦 ∧ 𝑦 ⊆ 𝐵 ) ) ) |
25 |
24
|
3ad2ant3 |
⊢ ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( 𝑥 ≼ 𝐵 ↔ ∃ 𝑦 ( 𝑥 ≈ 𝑦 ∧ 𝑦 ⊆ 𝐵 ) ) ) |
26 |
25
|
biimpa |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) → ∃ 𝑦 ( 𝑥 ≈ 𝑦 ∧ 𝑦 ⊆ 𝐵 ) ) |
27 |
|
simpl2 |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) → 𝑈 ∈ Univ ) |
28 |
|
gruss |
⊢ ( ( 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ 𝑦 ⊆ 𝐵 ) → 𝑦 ∈ 𝑈 ) |
29 |
28
|
3expia |
⊢ ( ( 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( 𝑦 ⊆ 𝐵 → 𝑦 ∈ 𝑈 ) ) |
30 |
29
|
3adant1 |
⊢ ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( 𝑦 ⊆ 𝐵 → 𝑦 ∈ 𝑈 ) ) |
31 |
30
|
adantr |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) → ( 𝑦 ⊆ 𝐵 → 𝑦 ∈ 𝑈 ) ) |
32 |
|
ensym |
⊢ ( 𝑥 ≈ 𝑦 → 𝑦 ≈ 𝑥 ) |
33 |
31 32
|
anim12d1 |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) → ( ( 𝑦 ⊆ 𝐵 ∧ 𝑥 ≈ 𝑦 ) → ( 𝑦 ∈ 𝑈 ∧ 𝑦 ≈ 𝑥 ) ) ) |
34 |
33
|
ancomsd |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) → ( ( 𝑥 ≈ 𝑦 ∧ 𝑦 ⊆ 𝐵 ) → ( 𝑦 ∈ 𝑈 ∧ 𝑦 ≈ 𝑥 ) ) ) |
35 |
34
|
eximdv |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) → ( ∃ 𝑦 ( 𝑥 ≈ 𝑦 ∧ 𝑦 ⊆ 𝐵 ) → ∃ 𝑦 ( 𝑦 ∈ 𝑈 ∧ 𝑦 ≈ 𝑥 ) ) ) |
36 |
|
gruen |
⊢ ( ( 𝑈 ∈ Univ ∧ 𝑥 ⊆ 𝑈 ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑦 ≈ 𝑥 ) ) → 𝑥 ∈ 𝑈 ) |
37 |
36
|
3com23 |
⊢ ( ( 𝑈 ∈ Univ ∧ ( 𝑦 ∈ 𝑈 ∧ 𝑦 ≈ 𝑥 ) ∧ 𝑥 ⊆ 𝑈 ) → 𝑥 ∈ 𝑈 ) |
38 |
37
|
3exp |
⊢ ( 𝑈 ∈ Univ → ( ( 𝑦 ∈ 𝑈 ∧ 𝑦 ≈ 𝑥 ) → ( 𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈 ) ) ) |
39 |
38
|
exlimdv |
⊢ ( 𝑈 ∈ Univ → ( ∃ 𝑦 ( 𝑦 ∈ 𝑈 ∧ 𝑦 ≈ 𝑥 ) → ( 𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈 ) ) ) |
40 |
27 35 39
|
sylsyld |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) → ( ∃ 𝑦 ( 𝑥 ≈ 𝑦 ∧ 𝑦 ⊆ 𝐵 ) → ( 𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈 ) ) ) |
41 |
26 40
|
mpd |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) → ( 𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈 ) ) |
42 |
23 41
|
syl5bir |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) → ( ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝑈 → 𝑥 ∈ 𝑈 ) ) |
43 |
22 42
|
syld |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) ∧ 𝑥 ≼ 𝐵 ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈 ) → 𝑥 ∈ 𝑈 ) ) |
44 |
43
|
ex |
⊢ ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( 𝑥 ≼ 𝐵 → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈 ) → 𝑥 ∈ 𝑈 ) ) ) |
45 |
44
|
com23 |
⊢ ( ( 𝑥 ∈ On ∧ 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈 ) → ( 𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈 ) ) ) |
46 |
45
|
3expib |
⊢ ( 𝑥 ∈ On → ( ( 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈 ) → ( 𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈 ) ) ) ) |
47 |
46
|
a2d |
⊢ ( 𝑥 ∈ On → ( ( ( 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ∀ 𝑦 ∈ 𝑥 ( 𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈 ) ) → ( ( 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( 𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈 ) ) ) ) |
48 |
9 47
|
syl5bi |
⊢ ( 𝑥 ∈ On → ( ∀ 𝑦 ∈ 𝑥 ( ( 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( 𝑦 ≼ 𝐵 → 𝑦 ∈ 𝑈 ) ) → ( ( 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( 𝑥 ≼ 𝐵 → 𝑥 ∈ 𝑈 ) ) ) ) |
49 |
4 8 48
|
tfis3 |
⊢ ( 𝐴 ∈ On → ( ( 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( 𝐴 ≼ 𝐵 → 𝐴 ∈ 𝑈 ) ) ) |
50 |
49
|
com3l |
⊢ ( ( 𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ) → ( 𝐴 ≼ 𝐵 → ( 𝐴 ∈ On → 𝐴 ∈ 𝑈 ) ) ) |
51 |
50
|
impr |
⊢ ( ( 𝑈 ∈ Univ ∧ ( 𝐵 ∈ 𝑈 ∧ 𝐴 ≼ 𝐵 ) ) → ( 𝐴 ∈ On → 𝐴 ∈ 𝑈 ) ) |
52 |
51
|
3impia |
⊢ ( ( 𝑈 ∈ Univ ∧ ( 𝐵 ∈ 𝑈 ∧ 𝐴 ≼ 𝐵 ) ∧ 𝐴 ∈ On ) → 𝐴 ∈ 𝑈 ) |
53 |
52
|
3com23 |
⊢ ( ( 𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ ( 𝐵 ∈ 𝑈 ∧ 𝐴 ≼ 𝐵 ) ) → 𝐴 ∈ 𝑈 ) |