Theorem elec 8325

Description: Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)

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		elec.2	. 2 ⊢ 𝐵 ∈ V
3		elecg 8324	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
4	1, 2, 3	mp2an 690	1 ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)

Syntax hints:   ↔ wb 208   ∈ wcel 2108  Vcvv 3493   class class class wbr 5057  [cec 8279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8283

This theorem is referenced by:  ecid  8354  sylow2alem2  18735  sylow2a  18736  sylow2blem1  18737  efgval2  18842  efgrelexlemb  18868  efgcpbllemb  18873  frgpnabllem1  18985  tgpconncomp  22713  qustgphaus  22723  vitalilem2  24202  vitalilem3  24203  isbndx  35052  prtlem10  35993  prtlem19  36006  prter3  36010