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Theorem ecexr 6434
 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 4885 . . . . 5 (𝐴 ∈ (𝑅 “ {𝐵}) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴))
21ibi 175 . . . 4 (𝐴 ∈ (𝑅 “ {𝐵}) → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴)
3 df-ec 6431 . . . 4 [𝐵]𝑅 = (𝑅 “ {𝐵})
42, 3eleq2s 2234 . . 3 (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴)
5 df-rex 2422 . . . 4 (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 ↔ ∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴))
6 simpl 108 . . . . . 6 ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 ∈ {𝐵})
7 velsn 3544 . . . . . 6 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
86, 7sylib 121 . . . . 5 ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 = 𝐵)
98eximi 1579 . . . 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 2692 . 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 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2417  Vcvv 2686  {csn 3527   class class class wbr 3929   “ cima 4542  [cec 6427 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-ec 6431 This theorem is referenced by:  relelec  6469  ecdmn0m  6471
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