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Metamath Proof Explorer

Theorem axprOLD

Description: Obsolete version of axpr as of 18-Sep-2025. (Contributed by NM, 14-Nov-2006) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion axprOLD ∃ 𝑧 ∀ 𝑤 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 )

Proof

Step Hyp Ref Expression
1 axprlem3OLD ⊢ ∃ 𝑧 ∀ 𝑤 ( 𝑤 ∈ 𝑧 ↔ ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) )
2 biimpr ⊢ ( ( 𝑤 ∈ 𝑧 ↔ ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) ) → ( ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) → 𝑤 ∈ 𝑧 ) )
3 2 alimi ⊢ ( ∀ 𝑤 ( 𝑤 ∈ 𝑧 ↔ ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) ) → ∀ 𝑤 ( ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) → 𝑤 ∈ 𝑧 ) )
4 1 3 eximii ⊢ ∃ 𝑧 ∀ 𝑤 ( ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) → 𝑤 ∈ 𝑧 )
5 axprlem4OLD ⊢ ( ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑥 ) → ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) )
6 axprlem5OLD ⊢ ( ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑦 ) → ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) )
7 5 6 jaodan ⊢ ( ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) ) → ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) )
8 7 ex ⊢ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) → ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) ) )
9 8 imim1d ⊢ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) → ( ( ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) → 𝑤 ∈ 𝑧 ) → ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) ) )
10 9 alimdv ⊢ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) → ( ∀ 𝑤 ( ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) → 𝑤 ∈ 𝑧 ) → ∀ 𝑤 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) ) )
11 10 eximdv ⊢ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) → ( ∃ 𝑧 ∀ 𝑤 ( ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) → 𝑤 ∈ 𝑧 ) → ∃ 𝑧 ∀ 𝑤 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) ) )
12 4 11 mpi ⊢ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) → ∃ 𝑧 ∀ 𝑤 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 ) )
13 axprlem2 ⊢ ∃ 𝑝 ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 )
14 12 13 exlimiiv ⊢ ∃ 𝑧 ∀ 𝑤 ( ( 𝑤 = 𝑥 ∨ 𝑤 = 𝑦 ) → 𝑤 ∈ 𝑧 )