Theorem axregszf

Description: Derivation of zfregs using ax-regs . (Contributed by BTernaryTau, 30-Dec-2025)

	Ref	Expression
Assertion	axregszf	⊢ ( 𝐴 ≠ ∅ → ∃ 𝑥 ∈ 𝐴 ( 𝑥 ∩ 𝐴 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		n0	⊢ ( 𝐴 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 )
2		axregscl	⊢ ( ∃ 𝑥 𝑥 ∈ 𝐴 → ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴 ) ) )
3		disj1	⊢ ( ( 𝑥 ∩ 𝐴 ) = ∅ ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴 ) )
4	3	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝑥 ∩ 𝐴 ) = ∅ ↔ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴 ) )
5		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴 ) ) )
6	4 5	bitr2i	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝐴 ) ) ↔ ∃ 𝑥 ∈ 𝐴 ( 𝑥 ∩ 𝐴 ) = ∅ )
7	2 6	sylib	⊢ ( ∃ 𝑥 𝑥 ∈ 𝐴 → ∃ 𝑥 ∈ 𝐴 ( 𝑥 ∩ 𝐴 ) = ∅ )
8	1 7	sylbi	⊢ ( 𝐴 ≠ ∅ → ∃ 𝑥 ∈ 𝐴 ( 𝑥 ∩ 𝐴 ) = ∅ )