Theorem n0abso

Description: Nonemptiness is absolute for transitive models. Compare Example I.16.3 of Kunen2 p. 96 and the following discussion. (Contributed by Eric Schmidt, 19-Oct-2025)

	Ref	Expression
Assertion	n0abso	⊢ ( ( Tr 𝑀 ∧ 𝐴 ∈ 𝑀 ) → ( 𝐴 ≠ ∅ ↔ ∃ 𝑥 ∈ 𝑀 𝑥 ∈ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		rexabso	⊢ ( ( Tr 𝑀 ∧ 𝐴 ∈ 𝑀 ) → ( ∃ 𝑥 ∈ 𝐴 ⊤ ↔ ∃ 𝑥 ∈ 𝑀 ( 𝑥 ∈ 𝐴 ∧ ⊤ ) ) )
2		tru	⊢ ⊤
3	2	rext0	⊢ ( ∃ 𝑥 ∈ 𝐴 ⊤ ↔ 𝐴 ≠ ∅ )
4	3	bicomi	⊢ ( 𝐴 ≠ ∅ ↔ ∃ 𝑥 ∈ 𝐴 ⊤ )
5	2	biantru	⊢ ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ ⊤ ) )
6	5	rexbii	⊢ ( ∃ 𝑥 ∈ 𝑀 𝑥 ∈ 𝐴 ↔ ∃ 𝑥 ∈ 𝑀 ( 𝑥 ∈ 𝐴 ∧ ⊤ ) )
7	1 4 6	3bitr4g	⊢ ( ( Tr 𝑀 ∧ 𝐴 ∈ 𝑀 ) → ( 𝐴 ≠ ∅ ↔ ∃ 𝑥 ∈ 𝑀 𝑥 ∈ 𝐴 ) )