Theorem ecexr 8650

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 4294	. . 3 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → ¬ (𝑅 “ {𝐵}) = ∅)
2		snprc 4676	. . . . 5 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
3		imaeq2 6023	. . . . 5 ⊢ ({𝐵} = ∅ → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
4	2, 3	sylbi 217	. . . 4 ⊢ (¬ 𝐵 ∈ V → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
5		ima0 6044	. . . 4 ⊢ (𝑅 “ ∅) = ∅
6	4, 5	eqtrdi 2788	. . 3 ⊢ (¬ 𝐵 ∈ V → (𝑅 “ {𝐵}) = ∅)
7	1, 6	nsyl2 141	. 2 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → 𝐵 ∈ V)
8		df-ec 8647	. 2 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵})
9	7, 8	eleq2s 2855	1 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287  {csn 4582   “ cima 5635  [cec 8643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ec 8647

This theorem is referenced by:  relelec  8693  ecdmn0  8698  erdisj  8703  eqvreldisj  38943