Theorem ecexr 8729

Description: A nonempty equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
ecexr	⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V)

Proof of Theorem ecexr

Step	Hyp	Ref	Expression
1		n0i 4320	. . 3 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → ¬ (𝑅 “ {𝐵}) = ∅)
2		snprc 4698	. . . . 5 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
3		imaeq2 6048	. . . . 5 ⊢ ({𝐵} = ∅ → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
4	2, 3	sylbi 217	. . . 4 ⊢ (¬ 𝐵 ∈ V → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
5		ima0 6069	. . . 4 ⊢ (𝑅 “ ∅) = ∅
6	4, 5	eqtrdi 2787	. . 3 ⊢ (¬ 𝐵 ∈ V → (𝑅 “ {𝐵}) = ∅)
7	1, 6	nsyl2 141	. 2 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → 𝐵 ∈ V)
8		df-ec 8726	. 2 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵})
9	7, 8	eleq2s 2853	1 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3464  ∅c0 4313  {csn 4606   “ cima 5662  [cec 8722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ec 8726

This theorem is referenced by:  relelec  8771  ecdmn0  8773  erdisj  8778  eqvreldisj  38637