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 485 | . . . 4 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → ∼ Er 𝑋) | |
2 | simprl 769 | . . . 4 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → 𝑥 ∈ (𝐴 / ∼ )) | |
3 | simprr 771 | . . . 4 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → 𝑦 ∈ (𝐴 / ∼ )) | |
4 | 1, 2, 3 | qsdisj 8368 | . . 3 ⊢ (( ∼ Er 𝑋 ∧ (𝑥 ∈ (𝐴 / ∼ ) ∧ 𝑦 ∈ (𝐴 / ∼ ))) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) |
5 | 4 | ralrimivva 3191 | . 2 ⊢ ( ∼ Er 𝑋 → ∀𝑥 ∈ (𝐴 / ∼ )∀𝑦 ∈ (𝐴 / ∼ )(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) |
6 | df-prt 36002 | . 2 ⊢ (Prt (𝐴 / ∼ ) ↔ ∀𝑥 ∈ (𝐴 / ∼ )∀𝑦 ∈ (𝐴 / ∼ )(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) | |
7 | 5, 6 | sylibr 236 | 1 ⊢ ( ∼ Er 𝑋 → Prt (𝐴 / ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∩ cin 3935 ∅c0 4291 Er wer 8280 / cqs 8282 Prt wprt 36001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-er 8283 df-ec 8285 df-qs 8289 df-prt 36002 |
This theorem is referenced by: (None) |
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