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Metamath Proof Explorer

Theorem el

Description: Any set is an element of some other set. See elALT for a shorter proof using more axioms, and see elALT2 for a proof that uses ax-9 and ax-pow instead of ax-pr . (Contributed by NM, 4-Jan-2002) (Proof shortened by Andrew Salmon, 25-Jul-2011) Use ax-pr instead of ax-9 and ax-pow . (Revised by BTernaryTau, 2-Dec-2024) (Proof shortened by Matthew House, 6-Apr-2026)

Ref Expression
Assertion el 
|- E. y x e. y

Proof

Step Hyp Ref Expression
1 ax-pr 
 |-  E. y A. z ( ( z = x \/ z = x ) -> z e. y )
2 orc 
 |-  ( z = x -> ( z = x \/ z = x ) )
3 ax8v1 
 |-  ( z = x -> ( z e. y -> x e. y ) )
4 2 3 embantd 
 |-  ( z = x -> ( ( ( z = x \/ z = x ) -> z e. y ) -> x e. y ) )
5 4 spimvw 
 |-  ( A. z ( ( z = x \/ z = x ) -> z e. y ) -> x e. y )
6 1 5 eximii 
 |-  E. y x e. y