Metamath Proof Explorer

Theorem axprlem5

Description: Lemma for axpr . The second element of the pair is included in any superset of the set whose existence is asserted by the axiom of replacement. (Contributed by Rohan Ridenour, 10-Aug-2023) (Revised by BJ, 13-Aug-2023)

Ref Expression
Assertion axprlem5 ( ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑦 ) → ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) )

Proof

Step Hyp Ref Expression
1 ax-nul ⊢ ∃ 𝑠 ∀ 𝑛 ¬ 𝑛 ∈ 𝑠
2 nfa1 ⊢ Ⅎ 𝑠 ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 )
3 nfv ⊢ Ⅎ 𝑠 𝑤 = 𝑦
4 2 3 nfan ⊢ Ⅎ 𝑠 ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑦 )
5 pm2.21 ⊢ ( ¬ 𝑛 ∈ 𝑠 → ( 𝑛 ∈ 𝑠 → ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) )
6 5 alimi ⊢ ( ∀ 𝑛 ¬ 𝑛 ∈ 𝑠 → ∀ 𝑛 ( 𝑛 ∈ 𝑠 → ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 ) )
7 6 adantr ⊢ ( ( ∀ 𝑛 ¬ 𝑛 ∈ 𝑠 ∧ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑦 ) ) → ∀ 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 ∧ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑦 ) ) → ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 )
10 sp ⊢ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) → ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) )
11 10 ad2antrl ⊢ ( ( ∀ 𝑛 ¬ 𝑛 ∈ 𝑠 ∧ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑦 ) ) → ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) )
12 9 11 mpd ⊢ ( ( ∀ 𝑛 ¬ 𝑛 ∈ 𝑠 ∧ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑦 ) ) → 𝑠 ∈ 𝑝 )
13 simpl ⊢ ( ( ∀ 𝑛 ¬ 𝑛 ∈ 𝑠 ∧ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑦 ) ) → ∀ 𝑛 ¬ 𝑛 ∈ 𝑠 )
14 alnex ⊢ ( ∀ 𝑛 ¬ 𝑛 ∈ 𝑠 ↔ ¬ ∃ 𝑛 𝑛 ∈ 𝑠 )
15 13 14 sylib ⊢ ( ( ∀ 𝑛 ¬ 𝑛 ∈ 𝑠 ∧ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑦 ) ) → ¬ ∃ 𝑛 𝑛 ∈ 𝑠 )
16 simprr ⊢ ( ( ∀ 𝑛 ¬ 𝑛 ∈ 𝑠 ∧ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑦 ) ) → 𝑤 = 𝑦 )
17 ifpfal ⊢ ( ¬ ∃ 𝑛 𝑛 ∈ 𝑠 → ( if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ↔ 𝑤 = 𝑦 ) )
18 17 biimpar ⊢ ( ( ¬ ∃ 𝑛 𝑛 ∈ 𝑠 ∧ 𝑤 = 𝑦 ) → if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) )
19 15 16 18 syl2anc ⊢ ( ( ∀ 𝑛 ¬ 𝑛 ∈ 𝑠 ∧ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑦 ) ) → if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) )
20 12 19 jca ⊢ ( ( ∀ 𝑛 ¬ 𝑛 ∈ 𝑠 ∧ ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑦 ) ) → ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) )
21 20 expcom ⊢ ( ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑦 ) → ( ∀ 𝑛 ¬ 𝑛 ∈ 𝑠 → ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) ) )
22 4 21 eximd ⊢ ( ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑦 ) → ( ∃ 𝑠 ∀ 𝑛 ¬ 𝑛 ∈ 𝑠 → ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) ) )
23 1 22 mpi ⊢ ( ( ∀ 𝑠 ( ∀ 𝑛 ∈ 𝑠 ∀ 𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝 ) ∧ 𝑤 = 𝑦 ) → ∃ 𝑠 ( 𝑠 ∈ 𝑝 ∧ if- ( ∃ 𝑛 𝑛 ∈ 𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) )