Step |
Hyp |
Ref |
Expression |
1 |
|
mnuprdlem4.1 |
⊢ 𝑀 = { 𝑘 ∣ ∀ 𝑙 ∈ 𝑘 ( 𝒫 𝑙 ⊆ 𝑘 ∧ ∀ 𝑚 ∃ 𝑛 ∈ 𝑘 ( 𝒫 𝑙 ⊆ 𝑛 ∧ ∀ 𝑝 ∈ 𝑙 ( ∃ 𝑞 ∈ 𝑘 ( 𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚 ) → ∃ 𝑟 ∈ 𝑚 ( 𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛 ) ) ) ) } |
2 |
|
mnuprdlem4.2 |
⊢ 𝐹 = { { ∅ , { 𝐴 } } , { { ∅ } , { 𝐵 } } } |
3 |
|
mnuprdlem4.3 |
⊢ ( 𝜑 → 𝑈 ∈ 𝑀 ) |
4 |
|
mnuprdlem4.4 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑈 ) |
5 |
|
mnuprdlem4.5 |
⊢ ( 𝜑 → 𝐵 ∈ 𝑈 ) |
6 |
|
mnuprdlem4.6 |
⊢ ( 𝜑 → ¬ 𝐴 = ∅ ) |
7 |
1 3 4
|
mnu0eld |
⊢ ( 𝜑 → ∅ ∈ 𝑈 ) |
8 |
1 3 7
|
mnusnd |
⊢ ( 𝜑 → { ∅ } ∈ 𝑈 ) |
9 |
|
0ss |
⊢ ∅ ⊆ { ∅ } |
10 |
|
ssid |
⊢ { ∅ } ⊆ { ∅ } |
11 |
1 3 8 9 10
|
mnuprss2d |
⊢ ( 𝜑 → { ∅ , { ∅ } } ∈ 𝑈 ) |
12 |
1 3 4
|
mnusnd |
⊢ ( 𝜑 → { 𝐴 } ∈ 𝑈 ) |
13 |
|
0ss |
⊢ ∅ ⊆ { 𝐴 } |
14 |
|
ssid |
⊢ { 𝐴 } ⊆ { 𝐴 } |
15 |
1 3 12 13 14
|
mnuprss2d |
⊢ ( 𝜑 → { ∅ , { 𝐴 } } ∈ 𝑈 ) |
16 |
|
0ss |
⊢ ∅ ⊆ 𝐵 |
17 |
|
ssid |
⊢ 𝐵 ⊆ 𝐵 |
18 |
1 3 5 16 17
|
mnuprss2d |
⊢ ( 𝜑 → { ∅ , 𝐵 } ∈ 𝑈 ) |
19 |
|
snsspr1 |
⊢ { ∅ } ⊆ { ∅ , 𝐵 } |
20 |
|
snsspr2 |
⊢ { 𝐵 } ⊆ { ∅ , 𝐵 } |
21 |
1 3 18 19 20
|
mnuprss2d |
⊢ ( 𝜑 → { { ∅ } , { 𝐵 } } ∈ 𝑈 ) |
22 |
15 21
|
prssd |
⊢ ( 𝜑 → { { ∅ , { 𝐴 } } , { { ∅ } , { 𝐵 } } } ⊆ 𝑈 ) |
23 |
2 22
|
eqsstrid |
⊢ ( 𝜑 → 𝐹 ⊆ 𝑈 ) |
24 |
1 3 11 23
|
mnuop3d |
⊢ ( 𝜑 → ∃ 𝑤 ∈ 𝑈 ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) |
25 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → 𝑤 ∈ 𝑈 ) |
26 |
|
eleq2w |
⊢ ( 𝑎 = 𝑤 → ( 𝐴 ∈ 𝑎 ↔ 𝐴 ∈ 𝑤 ) ) |
27 |
|
eleq2w |
⊢ ( 𝑎 = 𝑤 → ( 𝐵 ∈ 𝑎 ↔ 𝐵 ∈ 𝑤 ) ) |
28 |
26 27
|
anbi12d |
⊢ ( 𝑎 = 𝑤 → ( ( 𝐴 ∈ 𝑎 ∧ 𝐵 ∈ 𝑎 ) ↔ ( 𝐴 ∈ 𝑤 ∧ 𝐵 ∈ 𝑤 ) ) ) |
29 |
28
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ∧ 𝑎 = 𝑤 ) → ( ( 𝐴 ∈ 𝑎 ∧ 𝐵 ∈ 𝑎 ) ↔ ( 𝐴 ∈ 𝑤 ∧ 𝐵 ∈ 𝑤 ) ) ) |
30 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → 𝐴 ∈ 𝑈 ) |
31 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → 𝐵 ∈ 𝑈 ) |
32 |
|
nfv |
⊢ Ⅎ 𝑖 𝜑 |
33 |
|
nfv |
⊢ Ⅎ 𝑖 𝑤 ∈ 𝑈 |
34 |
|
nfra1 |
⊢ Ⅎ 𝑖 ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) |
35 |
33 34
|
nfan |
⊢ Ⅎ 𝑖 ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) |
36 |
32 35
|
nfan |
⊢ Ⅎ 𝑖 ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) |
37 |
2 36
|
mnuprdlem3 |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → ∀ 𝑖 ∈ { ∅ , { ∅ } } ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 ) |
38 |
|
ralim |
⊢ ( ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) → ( ∀ 𝑖 ∈ { ∅ , { ∅ } } ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∀ 𝑖 ∈ { ∅ , { ∅ } } ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) |
39 |
38
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → ( ∀ 𝑖 ∈ { ∅ , { ∅ } } ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∀ 𝑖 ∈ { ∅ , { ∅ } } ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) |
40 |
37 39
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → ∀ 𝑖 ∈ { ∅ , { ∅ } } ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) |
41 |
2 30 31 40
|
mnuprdlem1 |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → 𝐴 ∈ 𝑤 ) |
42 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → ¬ 𝐴 = ∅ ) |
43 |
2 31 42 40
|
mnuprdlem2 |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → 𝐵 ∈ 𝑤 ) |
44 |
41 43
|
jca |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → ( 𝐴 ∈ 𝑤 ∧ 𝐵 ∈ 𝑤 ) ) |
45 |
25 29 44
|
rspcedvd |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ 𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) → ∃ 𝑎 ∈ 𝑈 ( 𝐴 ∈ 𝑎 ∧ 𝐵 ∈ 𝑎 ) ) |
46 |
24 45
|
rexlimddv |
⊢ ( 𝜑 → ∃ 𝑎 ∈ 𝑈 ( 𝐴 ∈ 𝑎 ∧ 𝐵 ∈ 𝑎 ) ) |
47 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑈 ∧ ( 𝐴 ∈ 𝑎 ∧ 𝐵 ∈ 𝑎 ) ) ) → 𝑈 ∈ 𝑀 ) |
48 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑈 ∧ ( 𝐴 ∈ 𝑎 ∧ 𝐵 ∈ 𝑎 ) ) ) → 𝑎 ∈ 𝑈 ) |
49 |
|
simprrl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑈 ∧ ( 𝐴 ∈ 𝑎 ∧ 𝐵 ∈ 𝑎 ) ) ) → 𝐴 ∈ 𝑎 ) |
50 |
|
simprrr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑈 ∧ ( 𝐴 ∈ 𝑎 ∧ 𝐵 ∈ 𝑎 ) ) ) → 𝐵 ∈ 𝑎 ) |
51 |
49 50
|
prssd |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑈 ∧ ( 𝐴 ∈ 𝑎 ∧ 𝐵 ∈ 𝑎 ) ) ) → { 𝐴 , 𝐵 } ⊆ 𝑎 ) |
52 |
1 47 48 51
|
mnussd |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑈 ∧ ( 𝐴 ∈ 𝑎 ∧ 𝐵 ∈ 𝑎 ) ) ) → { 𝐴 , 𝐵 } ∈ 𝑈 ) |
53 |
46 52
|
rexlimddv |
⊢ ( 𝜑 → { 𝐴 , 𝐵 } ∈ 𝑈 ) |