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Metamath Proof Explorer

Theorem falseral0OLD

Description: Obsolete version of falseral0 as of 16-Feb-2026. (Contributed by AV, 30-Oct-2020) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion falseral0OLD ( ( ∀ 𝑥 ¬ 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝜑 ) → 𝐴 = ∅ )

Proof

Step Hyp Ref Expression
1 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
2 19.26 ⊢ ( ∀ 𝑥 ( ¬ 𝜑 ∧ ( 𝑥 ∈ 𝐴 → 𝜑 ) ) ↔ ( ∀ 𝑥 ¬ 𝜑 ∧ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )
3 con3 ⊢ ( ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( ¬ 𝜑 → ¬ 𝑥 ∈ 𝐴 ) )
4 3 impcom ⊢ ( ( ¬ 𝜑 ∧ ( 𝑥 ∈ 𝐴 → 𝜑 ) ) → ¬ 𝑥 ∈ 𝐴 )
5 4 alimi ⊢ ( ∀ 𝑥 ( ¬ 𝜑 ∧ ( 𝑥 ∈ 𝐴 → 𝜑 ) ) → ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )
6 alnex ⊢ ( ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ¬ ∃ 𝑥 𝑥 ∈ 𝐴 )
7 5 6 sylib ⊢ ( ∀ 𝑥 ( ¬ 𝜑 ∧ ( 𝑥 ∈ 𝐴 → 𝜑 ) ) → ¬ ∃ 𝑥 𝑥 ∈ 𝐴 )
8 notnotb ⊢ ( 𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅ )
9 neq0 ⊢ ( ¬ 𝐴 = ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 )
10 8 9 xchbinx ⊢ ( 𝐴 = ∅ ↔ ¬ ∃ 𝑥 𝑥 ∈ 𝐴 )
11 7 10 sylibr ⊢ ( ∀ 𝑥 ( ¬ 𝜑 ∧ ( 𝑥 ∈ 𝐴 → 𝜑 ) ) → 𝐴 = ∅ )
12 2 11 sylbir ⊢ ( ( ∀ 𝑥 ¬ 𝜑 ∧ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) → 𝐴 = ∅ )
13 1 12 sylan2b ⊢ ( ( ∀ 𝑥 ¬ 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝜑 ) → 𝐴 = ∅ )