Theorem clel5 3618

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.) Remove use of ax-10 2177, ax-11 2193, and ax-12 2212. (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 3237	. 2 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑋)
2		eqcom 2767	. . 3 ⊢ (𝑥 = 𝑋 ↔ 𝑋 = 𝑥)
3	2	rexbii 3109	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = 𝑋 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥)
4	1, 3	bitri 278	1 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2144  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752  df-clel 2835  df-rex 3087

This theorem is used by:  dfss5  4220  iunid  5018  4fvwrd4  13786  wrdlen1  14703  phisum  16974  symgmov1  19607  n0s0suc  28734  disjunsn  33236  rp-abid  44474