Theorem elec 3216

Description: Membership in an equivalence class. Theorem 72 of [Suppes] p. 82.

Hypotheses

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

Assertion

Ref	Expression
elec	⊢ (A ∈ [B]R ↔ BRA)

Proof of Theorem elec

Step	Hyp	Ref	Expression
1		elec.1	. 2 ⊢ A ∈ V
2		breq2 2066	. 2 ⊢ (x = A → (BRx ↔ BRA))
3		elec.2	. . 3 ⊢ B ∈ V
4	3	ec2 3203	. 2 ⊢ [B]R = {x∣BRx}
5	1, 2, 4	elab2 1419	1 ⊢ (A ∈ [B]R ↔ BRA)

Syntax hints:   ↔ wb 127   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054  [cec 3198

This theorem is referenced by:  ecdmn0 3217  erthi 3218  erth 3219  erdisj 3223

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-rex 1206  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-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-ec 3202

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