Theorem erprt 38866

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 770	. . . 4 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → 𝑥 ∈ (𝐴 / ∼ ))
3		simprr 772	. . . 4 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → 𝑦 ∈ (𝐴 / ∼ ))
4	1, 2, 3	qsdisj 8767	. . 3 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
5	4	ralrimivva 3180	. 2 ⊢ ( ∼ Er 𝑋 → ∀𝑥 ∈ (𝐴 / ∼ )∀𝑦 ∈ (𝐴 / ∼ )(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
6		df-prt 38865	. 2 ⊢ (Prt (𝐴 / ∼ ) ↔ ∀𝑥 ∈ (𝐴 / ∼ )∀𝑦 ∈ (𝐴 / ∼ )(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
7	5, 6	sylibr 234	1 ⊢ ( ∼ Er 𝑋 → Prt (𝐴 / ∼ ))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∩ cin 3913  ∅c0 4296   Er wer 8668   / cqs 8670  Prt wprt 38864

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-er 8671  df-ec 8673  df-qs 8677  df-prt 38865

This theorem is referenced by: (None)