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Metamath Proof Explorer

Theorem cardidd

Description: Any set is equinumerous to its cardinal number. Deduction form of cardid . (Contributed by David Moews, 1-May-2017)

Ref Expression
Hypothesis cardidd.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
Assertion cardidd ( 𝜑 → ( card ‘ 𝐴 ) ≈ 𝐴 )

Proof

Step Hyp Ref Expression
1 cardidd.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2 cardidg ⊢ ( 𝐴 ∈ 𝐵 → ( card ‘ 𝐴 ) ≈ 𝐴 )
3 1 2 syl ⊢ ( 𝜑 → ( card ‘ 𝐴 ) ≈ 𝐴 )