| Step |
Hyp |
Ref |
Expression |
| 1 |
|
naddcl |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +no 𝐵 ) ∈ On ) |
| 2 |
1
|
3adant3 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 +no 𝐵 ) ∈ On ) |
| 3 |
|
simp3 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → 𝐶 ∈ On ) |
| 4 |
|
naddov3 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +no 𝐵 ) = ∩ { 𝑎 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ∪ ( +no “ ( 𝐴 × { 𝐵 } ) ) ) ⊆ 𝑎 } ) |
| 5 |
4
|
3adant3 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 +no 𝐵 ) = ∩ { 𝑎 ∈ On ∣ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ∪ ( +no “ ( 𝐴 × { 𝐵 } ) ) ) ⊆ 𝑎 } ) |
| 6 |
|
intmin |
⊢ ( 𝐶 ∈ On → ∩ { 𝑐 ∈ On ∣ 𝐶 ⊆ 𝑐 } = 𝐶 ) |
| 7 |
6
|
eqcomd |
⊢ ( 𝐶 ∈ On → 𝐶 = ∩ { 𝑐 ∈ On ∣ 𝐶 ⊆ 𝑐 } ) |
| 8 |
7
|
3ad2ant3 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → 𝐶 = ∩ { 𝑐 ∈ On ∣ 𝐶 ⊆ 𝑐 } ) |
| 9 |
2 3 5 8
|
naddunif |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 +no 𝐵 ) +no 𝐶 ) = ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ∪ ( +no “ ( 𝐴 × { 𝐵 } ) ) ) × { 𝐶 } ) ) ∪ ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ) ⊆ 𝑥 } ) |
| 10 |
|
df-3an |
⊢ ( ( ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ⊆ 𝑥 ∧ ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ⊆ 𝑥 ∧ ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ⊆ 𝑥 ) ↔ ( ( ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ⊆ 𝑥 ∧ ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ⊆ 𝑥 ) ∧ ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ⊆ 𝑥 ) ) |
| 11 |
|
unss |
⊢ ( ( ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ⊆ 𝑥 ∧ ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ∪ ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ) ⊆ 𝑥 ) |
| 12 |
|
ancom |
⊢ ( ( ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ⊆ 𝑥 ∧ ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ⊆ 𝑥 ∧ ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ⊆ 𝑥 ) ) |
| 13 |
|
xpundir |
⊢ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ∪ ( +no “ ( 𝐴 × { 𝐵 } ) ) ) × { 𝐶 } ) = ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ∪ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) |
| 14 |
13
|
imaeq2i |
⊢ ( +no “ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ∪ ( +no “ ( 𝐴 × { 𝐵 } ) ) ) × { 𝐶 } ) ) = ( +no “ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ∪ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ) |
| 15 |
|
imaundi |
⊢ ( +no “ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ∪ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ) = ( ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ∪ ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ) |
| 16 |
14 15
|
eqtri |
⊢ ( +no “ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ∪ ( +no “ ( 𝐴 × { 𝐵 } ) ) ) × { 𝐶 } ) ) = ( ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ∪ ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ) |
| 17 |
16
|
sseq1i |
⊢ ( ( +no “ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ∪ ( +no “ ( 𝐴 × { 𝐵 } ) ) ) × { 𝐶 } ) ) ⊆ 𝑥 ↔ ( ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ∪ ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ) ⊆ 𝑥 ) |
| 18 |
11 12 17
|
3bitr4i |
⊢ ( ( ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ⊆ 𝑥 ∧ ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ⊆ 𝑥 ) ↔ ( +no “ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ∪ ( +no “ ( 𝐴 × { 𝐵 } ) ) ) × { 𝐶 } ) ) ⊆ 𝑥 ) |
| 19 |
18
|
anbi1i |
⊢ ( ( ( ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ⊆ 𝑥 ∧ ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ⊆ 𝑥 ) ∧ ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ∪ ( +no “ ( 𝐴 × { 𝐵 } ) ) ) × { 𝐶 } ) ) ⊆ 𝑥 ∧ ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ⊆ 𝑥 ) ) |
| 20 |
|
unss |
⊢ ( ( ( +no “ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ∪ ( +no “ ( 𝐴 × { 𝐵 } ) ) ) × { 𝐶 } ) ) ⊆ 𝑥 ∧ ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ⊆ 𝑥 ) ↔ ( ( +no “ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ∪ ( +no “ ( 𝐴 × { 𝐵 } ) ) ) × { 𝐶 } ) ) ∪ ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ) ⊆ 𝑥 ) |
| 21 |
10 19 20
|
3bitrri |
⊢ ( ( ( +no “ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ∪ ( +no “ ( 𝐴 × { 𝐵 } ) ) ) × { 𝐶 } ) ) ∪ ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ) ⊆ 𝑥 ↔ ( ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ⊆ 𝑥 ∧ ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ⊆ 𝑥 ∧ ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ⊆ 𝑥 ) ) |
| 22 |
|
naddfn |
⊢ +no Fn ( On × On ) |
| 23 |
|
fnfun |
⊢ ( +no Fn ( On × On ) → Fun +no ) |
| 24 |
22 23
|
ax-mp |
⊢ Fun +no |
| 25 |
|
imassrn |
⊢ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ ran +no |
| 26 |
|
naddf |
⊢ +no : ( On × On ) ⟶ On |
| 27 |
|
frn |
⊢ ( +no : ( On × On ) ⟶ On → ran +no ⊆ On ) |
| 28 |
26 27
|
ax-mp |
⊢ ran +no ⊆ On |
| 29 |
25 28
|
sstri |
⊢ ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ On |
| 30 |
|
simpl3 |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → 𝐶 ∈ On ) |
| 31 |
30
|
snssd |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → { 𝐶 } ⊆ On ) |
| 32 |
|
xpss12 |
⊢ ( ( ( +no “ ( 𝐴 × { 𝐵 } ) ) ⊆ On ∧ { 𝐶 } ⊆ On ) → ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ⊆ ( On × On ) ) |
| 33 |
29 31 32
|
sylancr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ⊆ ( On × On ) ) |
| 34 |
22
|
fndmi |
⊢ dom +no = ( On × On ) |
| 35 |
33 34
|
sseqtrrdi |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ⊆ dom +no ) |
| 36 |
|
funimassov |
⊢ ( ( Fun +no ∧ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ⊆ dom +no ) → ( ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ⊆ 𝑥 ↔ ∀ 𝑝 ∈ ( +no “ ( 𝐴 × { 𝐵 } ) ) ∀ 𝑐 ∈ { 𝐶 } ( 𝑝 +no 𝑐 ) ∈ 𝑥 ) ) |
| 37 |
24 35 36
|
sylancr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ⊆ 𝑥 ↔ ∀ 𝑝 ∈ ( +no “ ( 𝐴 × { 𝐵 } ) ) ∀ 𝑐 ∈ { 𝐶 } ( 𝑝 +no 𝑐 ) ∈ 𝑥 ) ) |
| 38 |
|
oveq2 |
⊢ ( 𝑐 = 𝐶 → ( 𝑝 +no 𝑐 ) = ( 𝑝 +no 𝐶 ) ) |
| 39 |
38
|
eleq1d |
⊢ ( 𝑐 = 𝐶 → ( ( 𝑝 +no 𝑐 ) ∈ 𝑥 ↔ ( 𝑝 +no 𝐶 ) ∈ 𝑥 ) ) |
| 40 |
39
|
ralsng |
⊢ ( 𝐶 ∈ On → ( ∀ 𝑐 ∈ { 𝐶 } ( 𝑝 +no 𝑐 ) ∈ 𝑥 ↔ ( 𝑝 +no 𝐶 ) ∈ 𝑥 ) ) |
| 41 |
30 40
|
syl |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ∀ 𝑐 ∈ { 𝐶 } ( 𝑝 +no 𝑐 ) ∈ 𝑥 ↔ ( 𝑝 +no 𝐶 ) ∈ 𝑥 ) ) |
| 42 |
41
|
ralbidv |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ∀ 𝑝 ∈ ( +no “ ( 𝐴 × { 𝐵 } ) ) ∀ 𝑐 ∈ { 𝐶 } ( 𝑝 +no 𝑐 ) ∈ 𝑥 ↔ ∀ 𝑝 ∈ ( +no “ ( 𝐴 × { 𝐵 } ) ) ( 𝑝 +no 𝐶 ) ∈ 𝑥 ) ) |
| 43 |
|
onss |
⊢ ( 𝐴 ∈ On → 𝐴 ⊆ On ) |
| 44 |
43
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → 𝐴 ⊆ On ) |
| 45 |
44
|
adantr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → 𝐴 ⊆ On ) |
| 46 |
|
simpl2 |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → 𝐵 ∈ On ) |
| 47 |
46
|
snssd |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → { 𝐵 } ⊆ On ) |
| 48 |
|
xpss12 |
⊢ ( ( 𝐴 ⊆ On ∧ { 𝐵 } ⊆ On ) → ( 𝐴 × { 𝐵 } ) ⊆ ( On × On ) ) |
| 49 |
45 47 48
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( 𝐴 × { 𝐵 } ) ⊆ ( On × On ) ) |
| 50 |
|
oveq1 |
⊢ ( 𝑝 = ( 𝑎 +no 𝑏 ) → ( 𝑝 +no 𝐶 ) = ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) ) |
| 51 |
50
|
eleq1d |
⊢ ( 𝑝 = ( 𝑎 +no 𝑏 ) → ( ( 𝑝 +no 𝐶 ) ∈ 𝑥 ↔ ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 52 |
51
|
imaeqalov |
⊢ ( ( +no Fn ( On × On ) ∧ ( 𝐴 × { 𝐵 } ) ⊆ ( On × On ) ) → ( ∀ 𝑝 ∈ ( +no “ ( 𝐴 × { 𝐵 } ) ) ( 𝑝 +no 𝐶 ) ∈ 𝑥 ↔ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ { 𝐵 } ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 53 |
22 49 52
|
sylancr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ∀ 𝑝 ∈ ( +no “ ( 𝐴 × { 𝐵 } ) ) ( 𝑝 +no 𝐶 ) ∈ 𝑥 ↔ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ { 𝐵 } ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 54 |
|
oveq2 |
⊢ ( 𝑏 = 𝐵 → ( 𝑎 +no 𝑏 ) = ( 𝑎 +no 𝐵 ) ) |
| 55 |
54
|
oveq1d |
⊢ ( 𝑏 = 𝐵 → ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) = ( ( 𝑎 +no 𝐵 ) +no 𝐶 ) ) |
| 56 |
55
|
eleq1d |
⊢ ( 𝑏 = 𝐵 → ( ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ↔ ( ( 𝑎 +no 𝐵 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 57 |
56
|
ralsng |
⊢ ( 𝐵 ∈ On → ( ∀ 𝑏 ∈ { 𝐵 } ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ↔ ( ( 𝑎 +no 𝐵 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 58 |
46 57
|
syl |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ∀ 𝑏 ∈ { 𝐵 } ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ↔ ( ( 𝑎 +no 𝐵 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 59 |
58
|
ralbidv |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ { 𝐵 } ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ↔ ∀ 𝑎 ∈ 𝐴 ( ( 𝑎 +no 𝐵 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 60 |
53 59
|
bitrd |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ∀ 𝑝 ∈ ( +no “ ( 𝐴 × { 𝐵 } ) ) ( 𝑝 +no 𝐶 ) ∈ 𝑥 ↔ ∀ 𝑎 ∈ 𝐴 ( ( 𝑎 +no 𝐵 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 61 |
37 42 60
|
3bitrd |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ⊆ 𝑥 ↔ ∀ 𝑎 ∈ 𝐴 ( ( 𝑎 +no 𝐵 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 62 |
|
imassrn |
⊢ ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ ran +no |
| 63 |
62 28
|
sstri |
⊢ ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ On |
| 64 |
|
xpss12 |
⊢ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ⊆ On ∧ { 𝐶 } ⊆ On ) → ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ⊆ ( On × On ) ) |
| 65 |
63 31 64
|
sylancr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ⊆ ( On × On ) ) |
| 66 |
65 34
|
sseqtrrdi |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ⊆ dom +no ) |
| 67 |
|
funimassov |
⊢ ( ( Fun +no ∧ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ⊆ dom +no ) → ( ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ⊆ 𝑥 ↔ ∀ 𝑝 ∈ ( +no “ ( { 𝐴 } × 𝐵 ) ) ∀ 𝑐 ∈ { 𝐶 } ( 𝑝 +no 𝑐 ) ∈ 𝑥 ) ) |
| 68 |
24 66 67
|
sylancr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ⊆ 𝑥 ↔ ∀ 𝑝 ∈ ( +no “ ( { 𝐴 } × 𝐵 ) ) ∀ 𝑐 ∈ { 𝐶 } ( 𝑝 +no 𝑐 ) ∈ 𝑥 ) ) |
| 69 |
41
|
ralbidv |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ∀ 𝑝 ∈ ( +no “ ( { 𝐴 } × 𝐵 ) ) ∀ 𝑐 ∈ { 𝐶 } ( 𝑝 +no 𝑐 ) ∈ 𝑥 ↔ ∀ 𝑝 ∈ ( +no “ ( { 𝐴 } × 𝐵 ) ) ( 𝑝 +no 𝐶 ) ∈ 𝑥 ) ) |
| 70 |
|
simpl1 |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → 𝐴 ∈ On ) |
| 71 |
70
|
snssd |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → { 𝐴 } ⊆ On ) |
| 72 |
|
onss |
⊢ ( 𝐵 ∈ On → 𝐵 ⊆ On ) |
| 73 |
72
|
3ad2ant2 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → 𝐵 ⊆ On ) |
| 74 |
73
|
adantr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → 𝐵 ⊆ On ) |
| 75 |
|
xpss12 |
⊢ ( ( { 𝐴 } ⊆ On ∧ 𝐵 ⊆ On ) → ( { 𝐴 } × 𝐵 ) ⊆ ( On × On ) ) |
| 76 |
71 74 75
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( { 𝐴 } × 𝐵 ) ⊆ ( On × On ) ) |
| 77 |
51
|
imaeqalov |
⊢ ( ( +no Fn ( On × On ) ∧ ( { 𝐴 } × 𝐵 ) ⊆ ( On × On ) ) → ( ∀ 𝑝 ∈ ( +no “ ( { 𝐴 } × 𝐵 ) ) ( 𝑝 +no 𝐶 ) ∈ 𝑥 ↔ ∀ 𝑎 ∈ { 𝐴 } ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 78 |
22 76 77
|
sylancr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ∀ 𝑝 ∈ ( +no “ ( { 𝐴 } × 𝐵 ) ) ( 𝑝 +no 𝐶 ) ∈ 𝑥 ↔ ∀ 𝑎 ∈ { 𝐴 } ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 79 |
|
oveq1 |
⊢ ( 𝑎 = 𝐴 → ( 𝑎 +no 𝑏 ) = ( 𝐴 +no 𝑏 ) ) |
| 80 |
79
|
oveq1d |
⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) = ( ( 𝐴 +no 𝑏 ) +no 𝐶 ) ) |
| 81 |
80
|
eleq1d |
⊢ ( 𝑎 = 𝐴 → ( ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ↔ ( ( 𝐴 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 82 |
81
|
ralbidv |
⊢ ( 𝑎 = 𝐴 → ( ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ↔ ∀ 𝑏 ∈ 𝐵 ( ( 𝐴 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 83 |
82
|
ralsng |
⊢ ( 𝐴 ∈ On → ( ∀ 𝑎 ∈ { 𝐴 } ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ↔ ∀ 𝑏 ∈ 𝐵 ( ( 𝐴 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 84 |
70 83
|
syl |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ∀ 𝑎 ∈ { 𝐴 } ∀ 𝑏 ∈ 𝐵 ( ( 𝑎 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ↔ ∀ 𝑏 ∈ 𝐵 ( ( 𝐴 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 85 |
78 84
|
bitrd |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ∀ 𝑝 ∈ ( +no “ ( { 𝐴 } × 𝐵 ) ) ( 𝑝 +no 𝐶 ) ∈ 𝑥 ↔ ∀ 𝑏 ∈ 𝐵 ( ( 𝐴 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 86 |
68 69 85
|
3bitrd |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ⊆ 𝑥 ↔ ∀ 𝑏 ∈ 𝐵 ( ( 𝐴 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ) ) |
| 87 |
2
|
adantr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( 𝐴 +no 𝐵 ) ∈ On ) |
| 88 |
87
|
snssd |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → { ( 𝐴 +no 𝐵 ) } ⊆ On ) |
| 89 |
|
onss |
⊢ ( 𝐶 ∈ On → 𝐶 ⊆ On ) |
| 90 |
89
|
3ad2ant3 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → 𝐶 ⊆ On ) |
| 91 |
90
|
adantr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → 𝐶 ⊆ On ) |
| 92 |
|
xpss12 |
⊢ ( ( { ( 𝐴 +no 𝐵 ) } ⊆ On ∧ 𝐶 ⊆ On ) → ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ⊆ ( On × On ) ) |
| 93 |
88 91 92
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ⊆ ( On × On ) ) |
| 94 |
93 34
|
sseqtrrdi |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ⊆ dom +no ) |
| 95 |
|
funimassov |
⊢ ( ( Fun +no ∧ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ⊆ dom +no ) → ( ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ⊆ 𝑥 ↔ ∀ 𝑎 ∈ { ( 𝐴 +no 𝐵 ) } ∀ 𝑐 ∈ 𝐶 ( 𝑎 +no 𝑐 ) ∈ 𝑥 ) ) |
| 96 |
24 94 95
|
sylancr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ⊆ 𝑥 ↔ ∀ 𝑎 ∈ { ( 𝐴 +no 𝐵 ) } ∀ 𝑐 ∈ 𝐶 ( 𝑎 +no 𝑐 ) ∈ 𝑥 ) ) |
| 97 |
|
ovex |
⊢ ( 𝐴 +no 𝐵 ) ∈ V |
| 98 |
|
oveq1 |
⊢ ( 𝑎 = ( 𝐴 +no 𝐵 ) → ( 𝑎 +no 𝑐 ) = ( ( 𝐴 +no 𝐵 ) +no 𝑐 ) ) |
| 99 |
98
|
eleq1d |
⊢ ( 𝑎 = ( 𝐴 +no 𝐵 ) → ( ( 𝑎 +no 𝑐 ) ∈ 𝑥 ↔ ( ( 𝐴 +no 𝐵 ) +no 𝑐 ) ∈ 𝑥 ) ) |
| 100 |
99
|
ralbidv |
⊢ ( 𝑎 = ( 𝐴 +no 𝐵 ) → ( ∀ 𝑐 ∈ 𝐶 ( 𝑎 +no 𝑐 ) ∈ 𝑥 ↔ ∀ 𝑐 ∈ 𝐶 ( ( 𝐴 +no 𝐵 ) +no 𝑐 ) ∈ 𝑥 ) ) |
| 101 |
97 100
|
ralsn |
⊢ ( ∀ 𝑎 ∈ { ( 𝐴 +no 𝐵 ) } ∀ 𝑐 ∈ 𝐶 ( 𝑎 +no 𝑐 ) ∈ 𝑥 ↔ ∀ 𝑐 ∈ 𝐶 ( ( 𝐴 +no 𝐵 ) +no 𝑐 ) ∈ 𝑥 ) |
| 102 |
96 101
|
bitrdi |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ⊆ 𝑥 ↔ ∀ 𝑐 ∈ 𝐶 ( ( 𝐴 +no 𝐵 ) +no 𝑐 ) ∈ 𝑥 ) ) |
| 103 |
61 86 102
|
3anbi123d |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ( ( +no “ ( ( +no “ ( 𝐴 × { 𝐵 } ) ) × { 𝐶 } ) ) ⊆ 𝑥 ∧ ( +no “ ( ( +no “ ( { 𝐴 } × 𝐵 ) ) × { 𝐶 } ) ) ⊆ 𝑥 ∧ ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ⊆ 𝑥 ) ↔ ( ∀ 𝑎 ∈ 𝐴 ( ( 𝑎 +no 𝐵 ) +no 𝐶 ) ∈ 𝑥 ∧ ∀ 𝑏 ∈ 𝐵 ( ( 𝐴 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ∧ ∀ 𝑐 ∈ 𝐶 ( ( 𝐴 +no 𝐵 ) +no 𝑐 ) ∈ 𝑥 ) ) ) |
| 104 |
21 103
|
bitrid |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ On ) → ( ( ( +no “ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ∪ ( +no “ ( 𝐴 × { 𝐵 } ) ) ) × { 𝐶 } ) ) ∪ ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ) ⊆ 𝑥 ↔ ( ∀ 𝑎 ∈ 𝐴 ( ( 𝑎 +no 𝐵 ) +no 𝐶 ) ∈ 𝑥 ∧ ∀ 𝑏 ∈ 𝐵 ( ( 𝐴 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ∧ ∀ 𝑐 ∈ 𝐶 ( ( 𝐴 +no 𝐵 ) +no 𝑐 ) ∈ 𝑥 ) ) ) |
| 105 |
104
|
rabbidva |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → { 𝑥 ∈ On ∣ ( ( +no “ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ∪ ( +no “ ( 𝐴 × { 𝐵 } ) ) ) × { 𝐶 } ) ) ∪ ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ) ⊆ 𝑥 } = { 𝑥 ∈ On ∣ ( ∀ 𝑎 ∈ 𝐴 ( ( 𝑎 +no 𝐵 ) +no 𝐶 ) ∈ 𝑥 ∧ ∀ 𝑏 ∈ 𝐵 ( ( 𝐴 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ∧ ∀ 𝑐 ∈ 𝐶 ( ( 𝐴 +no 𝐵 ) +no 𝑐 ) ∈ 𝑥 ) } ) |
| 106 |
105
|
inteqd |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ∩ { 𝑥 ∈ On ∣ ( ( +no “ ( ( ( +no “ ( { 𝐴 } × 𝐵 ) ) ∪ ( +no “ ( 𝐴 × { 𝐵 } ) ) ) × { 𝐶 } ) ) ∪ ( +no “ ( { ( 𝐴 +no 𝐵 ) } × 𝐶 ) ) ) ⊆ 𝑥 } = ∩ { 𝑥 ∈ On ∣ ( ∀ 𝑎 ∈ 𝐴 ( ( 𝑎 +no 𝐵 ) +no 𝐶 ) ∈ 𝑥 ∧ ∀ 𝑏 ∈ 𝐵 ( ( 𝐴 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ∧ ∀ 𝑐 ∈ 𝐶 ( ( 𝐴 +no 𝐵 ) +no 𝑐 ) ∈ 𝑥 ) } ) |
| 107 |
9 106
|
eqtrd |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 +no 𝐵 ) +no 𝐶 ) = ∩ { 𝑥 ∈ On ∣ ( ∀ 𝑎 ∈ 𝐴 ( ( 𝑎 +no 𝐵 ) +no 𝐶 ) ∈ 𝑥 ∧ ∀ 𝑏 ∈ 𝐵 ( ( 𝐴 +no 𝑏 ) +no 𝐶 ) ∈ 𝑥 ∧ ∀ 𝑐 ∈ 𝐶 ( ( 𝐴 +no 𝐵 ) +no 𝑐 ) ∈ 𝑥 ) } ) |