MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ecexr Structured version   Visualization version   GIF version

Theorem ecexr 7900
Description: A nonempty 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
StepHypRef Expression
1 n0i 4066 . . 3 (𝐴 ∈ (𝑅 “ {𝐵}) → ¬ (𝑅 “ {𝐵}) = ∅)
2 snprc 4387 . . . . 5 𝐵 ∈ V ↔ {𝐵} = ∅)
3 imaeq2 5603 . . . . 5 ({𝐵} = ∅ → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
42, 3sylbi 207 . . . 4 𝐵 ∈ V → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
5 ima0 5622 . . . 4 (𝑅 “ ∅) = ∅
64, 5syl6eq 2820 . . 3 𝐵 ∈ V → (𝑅 “ {𝐵}) = ∅)
71, 6nsyl2 144 . 2 (𝐴 ∈ (𝑅 “ {𝐵}) → 𝐵 ∈ V)
8 df-ec 7897 . 2 [𝐵]𝑅 = (𝑅 “ {𝐵})
97, 8eleq2s 2867 1 (𝐴 ∈ [𝐵]𝑅𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1630  wcel 2144  Vcvv 3349  c0 4061  {csn 4314  cima 5252  [cec 7893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ec 7897
This theorem is referenced by:  relelec  7938  ecdmn0  7940  erdisj  7945
  Copyright terms: Public domain W3C validator