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Mirrors > Home > MPE Home > Th. List > elqsn0 | Structured version Visualization version GIF version |
Description: A quotient set does not contain the empty set. (Contributed by NM, 24-Aug-1995.) |
Ref | Expression |
---|---|
elqsn0 | ⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . 2 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅) | |
2 | neeq1 3078 | . 2 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 ≠ ∅ ↔ 𝐵 ≠ ∅)) | |
3 | eleq2 2901 | . . . 4 ⊢ (dom 𝑅 = 𝐴 → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ 𝐴)) | |
4 | 3 | biimpar 480 | . . 3 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝑅) |
5 | ecdmn0 8330 | . . 3 ⊢ (𝑥 ∈ dom 𝑅 ↔ [𝑥]𝑅 ≠ ∅) | |
6 | 4, 5 | sylib 220 | . 2 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑥]𝑅 ≠ ∅) |
7 | 1, 2, 6 | ectocld 8358 | 1 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∅c0 4290 dom cdm 5549 [cec 8281 / cqs 8282 |
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 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
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 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ec 8285 df-qs 8289 |
This theorem is referenced by: ecelqsdm 8361 0nsr 10495 sylow1lem3 18719 vitalilem5 24207 prtlem400 36000 |
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