Theorem epelc 2123

Description: The epsilon relation and the membership relation are the same.

Hypotheses

Ref	Expression
epelc.1	⊢ A ∈ V
epelc.2	⊢ B ∈ V

Assertion

Ref	Expression
epelc	⊢ (AEB ↔ A ∈ B)

Proof of Theorem epelc

Step	Hyp	Ref	Expression
1		epelc.1	. 2 ⊢ A ∈ V
2		epelc.2	. 2 ⊢ B ∈ V
3		eleq1 1149	. 2 ⊢ (x = A → (x ∈ y ↔ A ∈ y))
4		eleq2 1150	. 2 ⊢ (y = B → (A ∈ y ↔ A ∈ B))
5		df-eprel 2122	. 2 ⊢ E = {⟨x, y⟩∣x ∈ y}
6	1, 2, 3, 4, 5	brab 2118	1 ⊢ (AEB ↔ A ∈ B)

Syntax hints:   ↔ wb 127   ∈ wel 803   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054  Ecep 2056

This theorem is referenced by:  epel 2124  ecid 3236  alephiso 3697

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-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-eprel 2122

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