Theorem prtex 38881

Description: The equivalence relation generated by a partition is a set if and only if the partition itself is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
prtlem18.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}

Assertion

Ref	Expression
prtex	⊢ (Prt 𝐴 → ( ∼ ∈ V ↔ 𝐴 ∈ V))

Distinct variable group:   𝑥,𝑢,𝑦,𝐴

Allowed substitution hints:   ∼ (𝑥,𝑦,𝑢)

Proof of Theorem prtex

Step	Hyp	Ref	Expression
1		prtlem18.1	. . . 4 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
2	1	prter1 38880	. . 3 ⊢ (Prt 𝐴 → ∼ Er ∪ 𝐴)
3		erexb 8770	. . 3 ⊢ ( ∼ Er ∪ 𝐴 → ( ∼ ∈ V ↔ ∪ 𝐴 ∈ V))
4	2, 3	syl 17	. 2 ⊢ (Prt 𝐴 → ( ∼ ∈ V ↔ ∪ 𝐴 ∈ V))
5		uniexb 7784	. 2 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
6	4, 5	bitr4di 289	1 ⊢ (Prt 𝐴 → ( ∼ ∈ V ↔ 𝐴 ∈ V))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  Vcvv 3480  ∪ cuni 4907  {copab 5205   Er wer 8742  Prt wprt 38872

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 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-er 8745  df-prt 38873

This theorem is referenced by: (None)