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Theorem ecexr 6564
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 4992 . . . . 5 (𝐴 ∈ (𝑅 “ {𝐵}) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴))
21ibi 176 . . . 4 (𝐴 ∈ (𝑅 “ {𝐵}) → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴)
3 df-ec 6561 . . . 4 [𝐵]𝑅 = (𝑅 “ {𝐵})
42, 3eleq2s 2284 . . 3 (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴)
5 df-rex 2474 . . . 4 (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 ↔ ∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴))
6 simpl 109 . . . . . 6 ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 ∈ {𝐵})
7 velsn 3624 . . . . . 6 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
86, 7sylib 122 . . . . 5 ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 = 𝐵)
98eximi 1611 . . . 4 (∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → ∃𝑥 𝑥 = 𝐵)
105, 9sylbi 121 . . 3 (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 → ∃𝑥 𝑥 = 𝐵)
114, 10syl 14 . 2 (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 𝑥 = 𝐵)
12 isset 2758 . 2 (𝐵 ∈ V ↔ ∃𝑥 𝑥 = 𝐵)
1311, 12sylibr 134 1 (𝐴 ∈ [𝐵]𝑅𝐵 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wex 1503  wcel 2160  wrex 2469  Vcvv 2752  {csn 3607   class class class wbr 4018  cima 4647  [cec 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-ec 6561
This theorem is referenced by:  relelec  6601  ecdmn0m  6603
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