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Metamath Proof Explorer

Theorem map1

Description: Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of Mendelson p. 255. (Contributed by NM, 17-Dec-2003) (Proof shortened by AV, 17-Jul-2022)

Ref Expression
Assertion map1 ( 𝐴 ∈ 𝑉 → ( 1o ↑m 𝐴 ) ≈ 1o )

Proof

Step Hyp Ref Expression
1 df1o2 ⊢ 1o = { ∅ }
2 1 oveq1i ⊢ ( 1o ↑m 𝐴 ) = ( { ∅ } ↑m 𝐴 )
3 0ex ⊢ ∅ ∈ V
4 snmapen1 ⊢ ( ( ∅ ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( { ∅ } ↑m 𝐴 ) ≈ 1o )
5 3 4 mpan ⊢ ( 𝐴 ∈ 𝑉 → ( { ∅ } ↑m 𝐴 ) ≈ 1o )
6 2 5 eqbrtrid ⊢ ( 𝐴 ∈ 𝑉 → ( 1o ↑m 𝐴 ) ≈ 1o )