Theorem clel5 3645

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 2129, ax-11 2146, and ax-12 2166. (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 3221	. 2 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝑋)
2		eqcom 2732	. . 3 ⊢ (𝑥 = 𝑋 ↔ 𝑋 = 𝑥)
3	2	rexbii 3084	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = 𝑋 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥)
4	1, 3	bitri 274	1 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥)

Syntax hints:   ↔ wb 205   = wceq 1533   ∈ wcel 2098  ∃wrex 3060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-cleq 2717  df-clel 2802  df-rex 3061

This theorem is referenced by:  dfss5  4257  iunid  5056  4fvwrd4  13651  wrdlen1  14534  phisum  16756  symgmov1  19343  disjunsn  32401  rp-abid  42844

Otomatik - 173.255.232.114 CloudFlare DNS Türk Telekom DNS Google DNS Open DNS

OSZAR »