Theorem ecexr 8288

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 4298	. . 3 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → ¬ (𝑅 “ {𝐵}) = ∅)
2		snprc 4646	. . . . 5 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
3		imaeq2 5919	. . . . 5 ⊢ ({𝐵} = ∅ → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
4	2, 3	sylbi 219	. . . 4 ⊢ (¬ 𝐵 ∈ V → (𝑅 “ {𝐵}) = (𝑅 “ ∅))
5		ima0 5939	. . . 4 ⊢ (𝑅 “ ∅) = ∅
6	4, 5	syl6eq 2872	. . 3 ⊢ (¬ 𝐵 ∈ V → (𝑅 “ {𝐵}) = ∅)
7	1, 6	nsyl2 143	. 2 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → 𝐵 ∈ V)
8		df-ec 8285	. 2 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵})
9	7, 8	eleq2s 2931	1 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1533   ∈ wcel 2110  Vcvv 3494  ∅c0 4290  {csn 4560   “ cima 5552  [cec 8281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-xp 5555  df-cnv 5557  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-ec 8285

This theorem is referenced by:  relelec  8328  ecdmn0  8330  erdisj  8335  eqvreldisj  35843