Theorem ecexr 8637

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 4293	. . 3 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → ¬ (𝑅 “ {𝐵}) = ∅)
2		snprc 4671	. . . . 5 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
3		imaeq2 6011	. . . . 5 ⊢ ({𝐵} = ∅ → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
4	2, 3	sylbi 217	. . . 4 ⊢ (¬ 𝐵 ∈ V → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
5		ima0 6032	. . . 4 ⊢ (𝑅 “ ∅) = ∅
6	4, 5	eqtrdi 2780	. . 3 ⊢ (¬ 𝐵 ∈ V → (𝑅 “ {𝐵}) = ∅)
7	1, 6	nsyl2 141	. 2 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → 𝐵 ∈ V)
8		df-ec 8634	. 2 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵})
9	7, 8	eleq2s 2846	1 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3438  ∅c0 4286  {csn 4579   “ cima 5626  [cec 8630

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 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ec 8634

This theorem is referenced by:  relelec  8679  ecdmn0  8684  erdisj  8689  eqvreldisj  38590