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Database ZF (ZERMELO-FRAENKEL) SET THEORY ZF Set Theory - start with the Axiom of Extensionality Restricted quantification Restricted universal and existential quantification raleleq
Description: All elements of a class are elements of a class equal to this class.
(Contributed by AV , 30-Oct-2020) (Proof shortened by Wolf Lammen , 18-Jul-2025)
Ref
Expression
Assertion
raleleq ⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 )
Proof
Step
Hyp
Ref
Expression
1
ralel ⊢ ∀ 𝑥 ∈ 𝐵 𝑥 ∈ 𝐵
2
raleq ⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐵 𝑥 ∈ 𝐵 ) )
3
1 2
mpbiri ⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 )