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Mirrors > Home > MPE Home > Th. List > qsdisj2 | Structured version Visualization version GIF version |
Description: A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.) |
Ref | Expression |
---|---|
qsdisj2 | ⊢ (𝑅 Er 𝑋 → Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . . . 4 ⊢ ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑅 Er 𝑋) | |
2 | simprl 769 | . . . 4 ⊢ ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑥 ∈ (𝐴 / 𝑅)) | |
3 | simprr 771 | . . . 4 ⊢ ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑦 ∈ (𝐴 / 𝑅)) | |
4 | 1, 2, 3 | qsdisj 8819 | . . 3 ⊢ ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) |
5 | 4 | ralrimivva 3198 | . 2 ⊢ (𝑅 Er 𝑋 → ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) |
6 | id 22 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
7 | 6 | disjor 5132 | . 2 ⊢ (Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥 ↔ ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) |
8 | 5, 7 | sylibr 233 | 1 ⊢ (𝑅 Er 𝑋 → Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ∩ cin 3948 ∅c0 4326 Disj wdisj 5117 Er wer 8728 / cqs 8730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-disj 5118 df-br 5153 df-opab 5215 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-er 8731 df-ec 8733 df-qs 8737 |
This theorem is referenced by: qshash 15813 |
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