Theorem prtex 39543

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 39542	. . 3 ⊢ (Prt 𝐴 → ∼ Er ∪ 𝐴)
3		erexb 8719	. . 3 ⊢ ( ∼ Er ∪ 𝐴 → ( ∼ ∈ V ↔ ∪ 𝐴 ∈ V))
4	2, 3	syl 18	. 2 ⊢ (Prt 𝐴 → ( ∼ ∈ V ↔ ∪ 𝐴 ∈ V))
5		uniexb 7762	. 2 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
6	4, 5	bitr4di 292	1 ⊢ (Prt 𝐴 → ( ∼ ∈ V ↔ 𝐴 ∈ V))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  Vcvv 3463  ∪ cuni 4876  {copab 5177   Er wer 8690  Prt wprt 39534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-er 8693  df-prt 39535

This theorem is referenced by: (None)