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Theorem ecexr 6313
Description: An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ecexr (𝐴 ∈ [𝐵]𝑅𝐵 ∈ V)

Proof of Theorem ecexr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elimag 4793 . . . . 5 (𝐴 ∈ (𝑅 “ {𝐵}) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴))
21ibi 175 . . . 4 (𝐴 ∈ (𝑅 “ {𝐵}) → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴)
3 df-ec 6310 . . . 4 [𝐵]𝑅 = (𝑅 “ {𝐵})
42, 3eleq2s 2183 . . 3 (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴)
5 df-rex 2366 . . . 4 (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 ↔ ∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴))
6 simpl 108 . . . . . 6 ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 ∈ {𝐵})
7 velsn 3469 . . . . . 6 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
86, 7sylib 121 . . . . 5 ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 = 𝐵)
98eximi 1537 . . . 4 (∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → ∃𝑥 𝑥 = 𝐵)
105, 9sylbi 120 . . 3 (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 → ∃𝑥 𝑥 = 𝐵)
114, 10syl 14 . 2 (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 𝑥 = 𝐵)
12 isset 2628 . 2 (𝐵 ∈ V ↔ ∃𝑥 𝑥 = 𝐵)
1311, 12sylibr 133 1 (𝐴 ∈ [𝐵]𝑅𝐵 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1290  wex 1427  wcel 1439  wrex 2361  Vcvv 2622  {csn 3452   class class class wbr 3853  cima 4457  [cec 6306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-opab 3908  df-xp 4460  df-cnv 4462  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-ec 6310
This theorem is referenced by:  relelec  6348  ecdmn0m  6350
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