Theorem clel5 40303

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.)

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		id 22	. . 3 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ 𝐴)
2		eqeq2 2621	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑋 = 𝑥 ↔ 𝑋 = 𝑋))
3	2	adantl 481	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) → (𝑋 = 𝑥 ↔ 𝑋 = 𝑋))
4		eqidd 2611	. . 3 ⊢ (𝑋 ∈ 𝐴 → 𝑋 = 𝑋)
5	1, 3, 4	rspcedvd 3289	. 2 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝑋 = 𝑥)
6		eleq1a 2683	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑋 = 𝑥 → 𝑋 ∈ 𝐴))
7	6	rexlimiv 3009	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑋 = 𝑥 → 𝑋 ∈ 𝐴)
8	5, 7	impbii 198	1 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥)

Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175

This theorem is referenced by:  dfss7  40304