Theorem risset 1235

Description: Two ways to say "A belongs to B".

Assertion

Ref	Expression
risset	⊢ (A ∈ B ↔ ∃x ∈ B x = A)

Distinct variable group(s):   x,A   x,B

Proof of Theorem risset

Step	Hyp	Ref	Expression
1		exancom 736	. 2 ⊢ (∃x(x ∈ B ∧ x = A) ↔ ∃x(x = A ∧ x ∈ B))
2		df-rex 1206	. 2 ⊢ (∃x ∈ B x = A ↔ ∃x(x ∈ B ∧ x = A))
3		df-clel 1099	. 2 ⊢ (A ∈ B ↔ ∃x(x = A ∧ x ∈ B))
4	1, 2, 3	3bitr4r 159	1 ⊢ (A ∈ B ↔ ∃x ∈ B x = A)

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  0el 1720  sucel 2296  qsid 3237  zorn2 3612  negeu 4124  receu 4215  zqt 4632

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-clel 1099  df-rex 1206

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