Metamath Proof Explorer


Theorem mnuprdlem4

Description: Lemma for mnuprd . General case. (Contributed by Rohan Ridenour, 13-Aug-2023)

Ref Expression
Hypotheses mnuprdlem4.1 𝑀 = { 𝑘 ∣ ∀ 𝑙𝑘 ( 𝒫 𝑙𝑘 ∧ ∀ 𝑚𝑛𝑘 ( 𝒫 𝑙𝑛 ∧ ∀ 𝑝𝑙 ( ∃ 𝑞𝑘 ( 𝑝𝑞𝑞𝑚 ) → ∃ 𝑟𝑚 ( 𝑝𝑟 𝑟𝑛 ) ) ) ) }
mnuprdlem4.2 𝐹 = { { ∅ , { 𝐴 } } , { { ∅ } , { 𝐵 } } }
mnuprdlem4.3 ( 𝜑𝑈𝑀 )
mnuprdlem4.4 ( 𝜑𝐴𝑈 )
mnuprdlem4.5 ( 𝜑𝐵𝑈 )
mnuprdlem4.6 ( 𝜑 → ¬ 𝐴 = ∅ )
Assertion mnuprdlem4 ( 𝜑 → { 𝐴 , 𝐵 } ∈ 𝑈 )

Proof

Step Hyp Ref Expression
1 mnuprdlem4.1 𝑀 = { 𝑘 ∣ ∀ 𝑙𝑘 ( 𝒫 𝑙𝑘 ∧ ∀ 𝑚𝑛𝑘 ( 𝒫 𝑙𝑛 ∧ ∀ 𝑝𝑙 ( ∃ 𝑞𝑘 ( 𝑝𝑞𝑞𝑚 ) → ∃ 𝑟𝑚 ( 𝑝𝑟 𝑟𝑛 ) ) ) ) }
2 mnuprdlem4.2 𝐹 = { { ∅ , { 𝐴 } } , { { ∅ } , { 𝐵 } } }
3 mnuprdlem4.3 ( 𝜑𝑈𝑀 )
4 mnuprdlem4.4 ( 𝜑𝐴𝑈 )
5 mnuprdlem4.5 ( 𝜑𝐵𝑈 )
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 𝐵𝐵
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 ( 𝜑𝐹𝑈 )
24 1 3 11 23 mnuop3d ( 𝜑 → ∃ 𝑤𝑈𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) )
25 simprl ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → 𝑤𝑈 )
26 eleq2w ( 𝑎 = 𝑤 → ( 𝐴𝑎𝐴𝑤 ) )
27 eleq2w ( 𝑎 = 𝑤 → ( 𝐵𝑎𝐵𝑤 ) )
28 26 27 anbi12d ( 𝑎 = 𝑤 → ( ( 𝐴𝑎𝐵𝑎 ) ↔ ( 𝐴𝑤𝐵𝑤 ) ) )
29 28 adantl ( ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) ) ) ∧ 𝑎 = 𝑤 ) → ( ( 𝐴𝑎𝐵𝑎 ) ↔ ( 𝐴𝑤𝐵𝑤 ) ) )
30 4 adantr ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → 𝐴𝑈 )
31 5 adantr ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → 𝐵𝑈 )
32 nfv 𝑖 𝜑
33 nfv 𝑖 𝑤𝑈
34 nfra1 𝑖𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) )
35 33 34 nfan 𝑖 ( 𝑤𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) )
36 32 35 nfan 𝑖 ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) ) )
37 2 36 mnuprdlem3 ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → ∀ 𝑖 ∈ { ∅ , { ∅ } } ∃ 𝑣𝐹 𝑖𝑣 )
38 ralim ( ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) → ( ∀ 𝑖 ∈ { ∅ , { ∅ } } ∃ 𝑣𝐹 𝑖𝑣 → ∀ 𝑖 ∈ { ∅ , { ∅ } } ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) )
39 38 ad2antll ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → ( ∀ 𝑖 ∈ { ∅ , { ∅ } } ∃ 𝑣𝐹 𝑖𝑣 → ∀ 𝑖 ∈ { ∅ , { ∅ } } ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) )
40 37 39 mpd ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → ∀ 𝑖 ∈ { ∅ , { ∅ } } ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) )
41 2 30 31 40 mnuprdlem1 ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → 𝐴𝑤 )
42 6 adantr ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → ¬ 𝐴 = ∅ )
43 2 31 42 40 mnuprdlem2 ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → 𝐵𝑤 )
44 41 43 jca ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → ( 𝐴𝑤𝐵𝑤 ) )
45 25 29 44 rspcedvd ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖 ∈ { ∅ , { ∅ } } ( ∃ 𝑣𝐹 𝑖𝑣 → ∃ 𝑢𝐹 ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → ∃ 𝑎𝑈 ( 𝐴𝑎𝐵𝑎 ) )
46 24 45 rexlimddv ( 𝜑 → ∃ 𝑎𝑈 ( 𝐴𝑎𝐵𝑎 ) )
47 3 adantr ( ( 𝜑 ∧ ( 𝑎𝑈 ∧ ( 𝐴𝑎𝐵𝑎 ) ) ) → 𝑈𝑀 )
48 simprl ( ( 𝜑 ∧ ( 𝑎𝑈 ∧ ( 𝐴𝑎𝐵𝑎 ) ) ) → 𝑎𝑈 )
49 simprrl ( ( 𝜑 ∧ ( 𝑎𝑈 ∧ ( 𝐴𝑎𝐵𝑎 ) ) ) → 𝐴𝑎 )
50 simprrr ( ( 𝜑 ∧ ( 𝑎𝑈 ∧ ( 𝐴𝑎𝐵𝑎 ) ) ) → 𝐵𝑎 )
51 49 50 prssd ( ( 𝜑 ∧ ( 𝑎𝑈 ∧ ( 𝐴𝑎𝐵𝑎 ) ) ) → { 𝐴 , 𝐵 } ⊆ 𝑎 )
52 1 47 48 51 mnussd ( ( 𝜑 ∧ ( 𝑎𝑈 ∧ ( 𝐴𝑎𝐵𝑎 ) ) ) → { 𝐴 , 𝐵 } ∈ 𝑈 )
53 46 52 rexlimddv ( 𝜑 → { 𝐴 , 𝐵 } ∈ 𝑈 )