Theorem erprt 39380

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 484	. . . 4 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → ∼ Er 𝑋)
2		simprl 777	. . . 4 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → 𝑥 ∈ (𝐴 / ∼ ))
3		simprr 779	. . . 4 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → 𝑦 ∈ (𝐴 / ∼ ))
4	1, 2, 3	qsdisj 8735	. . 3 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
5	4	ralrimivva 3184	. 2 ⊢ ( ∼ Er 𝑋 → ∀𝑥 ∈ (𝐴 / ∼ )∀𝑦 ∈ (𝐴 / ∼ )(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
6		df-prt 39379	. 2 ⊢ (Prt (𝐴 / ∼ ) ↔ ∀𝑥 ∈ (𝐴 / ∼ )∀𝑦 ∈ (𝐴 / ∼ )(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
7	5, 6	sylibr 236	1 ⊢ ( ∼ Er 𝑋 → Prt (𝐴 / ∼ ))

Syntax hints:   → wi 4   ∧ wa 397   ∨ wo 854   = wceq 1548   ∈ wcel 2121  ∀wral 3055   ∩ cin 3884  ∅c0 4264   Er wer 8634   / cqs 8636  Prt wprt 39378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-er 8637  df-ec 8639  df-qs 8643  df-prt 39379

This theorem is referenced by: (None)