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Theorem elecg 6172
Description: Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
elecg ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))

Proof of Theorem elecg
StepHypRef Expression
1 elimasng 4718 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
21ancoms 259 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
3 df-ec 6136 . . 3 [𝐵]𝑅 = (𝑅 “ {𝐵})
43eleq2i 2118 . 2 (𝐴 ∈ [𝐵]𝑅𝐴 ∈ (𝑅 “ {𝐵}))
5 df-br 3790 . 2 (𝐵𝑅𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
62, 4, 53bitr4g 216 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wcel 1407  {csn 3400  cop 3403   class class class wbr 3789  cima 4373  [cec 6132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-sbc 2785  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790  df-opab 3844  df-xp 4376  df-cnv 4378  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-ec 6136
This theorem is referenced by:  elec  6173  relelec  6174  ecdmn0m  6176  erth  6178  ecidg  6198  qsel  6211
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