Theorem erprt 36814

Description: The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)

Assertion

Ref	Expression
erprt	⊢ ( ∼ Er 𝑋 → Prt (𝐴 / ∼ ))

Proof of Theorem erprt

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → ∼ Er 𝑋)
2		simprl 767	. . . 4 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → 𝑥 ∈ (𝐴 / ∼ ))
3		simprr 769	. . . 4 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → 𝑦 ∈ (𝐴 / ∼ ))
4	1, 2, 3	qsdisj 8541	. . 3 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
5	4	ralrimivva 3114	. 2 ⊢ ( ∼ Er 𝑋 → ∀𝑥 ∈ (𝐴 / ∼ )∀𝑦 ∈ (𝐴 / ∼ )(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
6		df-prt 36813	. 2 ⊢ (Prt (𝐴 / ∼ ) ↔ ∀𝑥 ∈ (𝐴 / ∼ )∀𝑦 ∈ (𝐴 / ∼ )(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
7	5, 6	sylibr 233	1 ⊢ ( ∼ Er 𝑋 → Prt (𝐴 / ∼ ))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ∩ cin 3882  ∅c0 4253   Er wer 8453   / cqs 8455  Prt wprt 36812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-er 8456  df-ec 8458  df-qs 8462  df-prt 36813

This theorem is referenced by: (None)