Theorem ecexr 8633

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 4289	. . 3 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → ¬ (𝑅 “ {𝐵}) = ∅)
2		snprc 4669	. . . . 5 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
3		imaeq2 6009	. . . . 5 ⊢ ({𝐵} = ∅ → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
4	2, 3	sylbi 217	. . . 4 ⊢ (¬ 𝐵 ∈ V → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
5		ima0 6030	. . . 4 ⊢ (𝑅 “ ∅) = ∅
6	4, 5	eqtrdi 2784	. . 3 ⊢ (¬ 𝐵 ∈ V → (𝑅 “ {𝐵}) = ∅)
7	1, 6	nsyl2 141	. 2 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → 𝐵 ∈ V)
8		df-ec 8630	. 2 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵})
9	7, 8	eleq2s 2851	1 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ∅c0 4282  {csn 4575   “ cima 5622  [cec 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ec 8630

This theorem is referenced by:  relelec  8675  ecdmn0  8680  erdisj  8685  eqvreldisj  38730