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Metamath Proof Explorer

Theorem kmlem7

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004)

Ref Expression
Assertion kmlem7 ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) → ¬ ∃ 𝑧 ∈ 𝑥 ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) )

Proof

Step Hyp Ref Expression
1 kmlem6 ⊢ ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) → ∀ 𝑧 ∈ 𝑥 ∃ 𝑣 ∈ 𝑧 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) )
2 ralinexa ⊢ ( ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) ↔ ¬ ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) )
3 2 rexbii ⊢ ( ∃ 𝑣 ∈ 𝑧 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) ↔ ∃ 𝑣 ∈ 𝑧 ¬ ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) )
4 rexnal ⊢ ( ∃ 𝑣 ∈ 𝑧 ¬ ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) ↔ ¬ ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) )
5 3 4 bitri ⊢ ( ∃ 𝑣 ∈ 𝑧 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) ↔ ¬ ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) )
6 5 ralbii ⊢ ( ∀ 𝑧 ∈ 𝑥 ∃ 𝑣 ∈ 𝑧 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) ↔ ∀ 𝑧 ∈ 𝑥 ¬ ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) )
7 ralnex ⊢ ( ∀ 𝑧 ∈ 𝑥 ¬ ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) ↔ ¬ ∃ 𝑧 ∈ 𝑥 ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) )
8 6 7 bitri ⊢ ( ∀ 𝑧 ∈ 𝑥 ∃ 𝑣 ∈ 𝑧 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) ↔ ¬ ∃ 𝑧 ∈ 𝑥 ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) )
9 1 8 sylib ⊢ ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) → ¬ ∃ 𝑧 ∈ 𝑥 ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) )