Theorem clel5 2897

Description: Alternate definition of class membership: a class 𝑋 is an element of another class 𝐴 iff there is an element of 𝐴 equal to 𝑋. (Contributed by AV, 13-Nov-2020.) (Revised by Steven Nguyen, 19-May-2023.)

Assertion

Ref	Expression
clel5	⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Proof of Theorem clel5

Step	Hyp	Ref	Expression
1		risset 2522	. 2 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑋)
2		eqcom 2195	. . 3 ⊢ (𝑥 = 𝑋 ↔ 𝑋 = 𝑥)
3	2	rexbii 2501	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = 𝑋 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥)
4	1, 3	bitri 184	1 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥)

Syntax hints:   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∃wrex 2473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-cleq 2186  df-clel 2189  df-rex 2478

This theorem is referenced by:  wrdlen1  10951  phisum  12378