Theorem clel5 3343

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 2633	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑋 = 𝑥 ↔ 𝑋 = 𝑋))
3	2	adantl 482	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) → (𝑋 = 𝑥 ↔ 𝑋 = 𝑋))
4		eqidd 2623	. . 3 ⊢ (𝑋 ∈ 𝐴 → 𝑋 = 𝑋)
5	1, 3, 4	rspcedvd 3317	. 2 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝑋 = 𝑥)
6		eleq1a 2696	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑋 = 𝑥 → 𝑋 ∈ 𝐴))
7	6	rexlimiv 3027	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑋 = 𝑥 → 𝑋 ∈ 𝐴)
8	5, 7	impbii 199	1 ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥)

Syntax hints:   ↔ wb 196   = wceq 1483   ∈ wcel 1990  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202

This theorem is referenced by:  dfss5  3864  disjunsn  29407