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Mathbox for Rodolfo Medina |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > erprt | Structured version Visualization version GIF version |
Description: The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
Ref | Expression |
---|---|
erprt | ⊢ ( ∼ Er 𝑋 → Prt (𝐴 / ∼ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . . . 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 8739 | . . 3 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) |
5 | 4 | ralrimivva 3194 | . 2 ⊢ ( ∼ Er 𝑋 → ∀𝑥 ∈ (𝐴 / ∼ )∀𝑦 ∈ (𝐴 / ∼ )(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) |
6 | df-prt 37384 | . 2 ⊢ (Prt (𝐴 / ∼ ) ↔ ∀𝑥 ∈ (𝐴 / ∼ )∀𝑦 ∈ (𝐴 / ∼ )(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) | |
7 | 5, 6 | sylibr 233 | 1 ⊢ ( ∼ Er 𝑋 → Prt (𝐴 / ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∩ cin 3913 ∅c0 4286 Er wer 8651 / cqs 8653 Prt wprt 37383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-er 8654 df-ec 8656 df-qs 8660 df-prt 37384 |
This theorem is referenced by: (None) |
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