This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem axprlem4OLD

Description: Obsolete version of axprlem4 as of 18-Sep-2025. (Contributed by Rohan Ridenour, 10-Aug-2023) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion axprlem4OLD ( ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑥 ) → ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) )

Proof

Step Hyp Ref Expression
1 axprlem1 ⊢ ∃ 𝑠 ∀ 𝑛 ( ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑛 ∈ 𝑠 )
2 1 bm1.3iiOLD ⊢ ∃ 𝑠 ∀ 𝑛 ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 )
3 nfa1 ⊢ Ⅎ 𝑠 ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 )
4 nfv ⊢ Ⅎ 𝑠 𝑤 = 𝑥
5 3 4 nfan ⊢ Ⅎ 𝑠 ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑥 )
6 biimp ⊢ ( ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) → ( 𝑛 ∈ 𝑠 → ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) )
7 6 alimi ⊢ ( ∀ 𝑛 ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) → ∀ 𝑛 ( 𝑛 ∈ 𝑠 → ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) )
8 df-ral ⊢ ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ↔ ∀ 𝑛 ( 𝑛 ∈ 𝑠 → ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) )
9 7 8 sylibr ⊢ ( ∀ 𝑛 ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) → ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 )
10 sp ⊢ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) → ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) )
11 9 10 mpan9 ⊢ ( ( ∀ 𝑛 ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) ∧ ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ) → 𝑠 ∈ 𝑝 )
12 11 adantrr ⊢ ( ( ∀ 𝑛 ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) ∧ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑥 ) ) → 𝑠 ∈ 𝑝 )
13 ax-nul ⊢ ∃ 𝑛 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛
14 nfa1 ⊢ Ⅎ 𝑛 ∀ 𝑛 ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 )
15 sp ⊢ ( ∀ 𝑛 ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) → ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) )
16 15 biimprd ⊢ ( ∀ 𝑛 ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) → ( ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑛 ∈ 𝑠 ) )
17 14 16 eximd ⊢ ( ∀ 𝑛 ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) → ( ∃ 𝑛 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → ∃ 𝑛 𝑛 ∈ 𝑠 ) )
18 13 17 mpi ⊢ ( ∀ 𝑛 ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) → ∃ 𝑛 𝑛 ∈ 𝑠 )
19 simprr ⊢ ( ( ∀ 𝑛 ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) ∧ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑥 ) ) → 𝑤 = 𝑥 )
20 ifptru ⊢ ( ∃ 𝑛 𝑛 ∈ 𝑠 → ( if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ↔ 𝑤 = 𝑥 ) )
21 20 biimpar ⊢ ( ( ∃ 𝑛 𝑛 ∈ 𝑠 ∧ 𝑤 = 𝑥 ) → if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) )
22 18 19 21 syl2an2r ⊢ ( ( ∀ 𝑛 ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) ∧ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑥 ) ) → if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) )
23 12 22 jca ⊢ ( ( ∀ 𝑛 ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) ∧ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑥 ) ) → ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) )
24 23 expcom ⊢ ( ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑥 ) → ( ∀ 𝑛 ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) → ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) ) )
25 5 24 eximd ⊢ ( ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑥 ) → ( ∃ 𝑠 ∀ 𝑛 ( 𝑛 ∈ 𝑠 ↔ ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) → ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) ) )
26 2 25 mpi ⊢ ( ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑥 ) → ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) )