Metamath Proof Explorer


Theorem dif1ennn

Description: If a set A is equinumerous to the successor of a natural number M , then A with an element removed is equinumerous to M . See also dif1ennnALT . (Contributed by BTernaryTau, 6-Jan-2025)

Ref Expression
Assertion dif1ennn ( ( 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴 ) → ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑀 )

Proof

Step Hyp Ref Expression
1 nnon ( 𝑀 ∈ ω → 𝑀 ∈ On )
2 dif1en ( ( 𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀𝑋𝐴 ) → ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑀 )
3 1 2 syl3an1 ( ( 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴 ) → ( 𝐴 ∖ { 𝑋 } ) ≈ 𝑀 )