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Mirrors > Home > MPE Home > Th. List > elqsn0 | Structured version Visualization version GIF version |
Description: A quotient set doesn't 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 2610 | . 2 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅) | |
2 | neeq1 2844 | . 2 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 ≠ ∅ ↔ 𝐵 ≠ ∅)) | |
3 | eleq2 2677 | . . . 4 ⊢ (dom 𝑅 = 𝐴 → (𝑥 ∈ dom 𝑅 ↔ 𝑥 ∈ 𝐴)) | |
4 | 3 | biimpar 501 | . . 3 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝑅) |
5 | ecdmn0 7676 | . . 3 ⊢ (𝑥 ∈ dom 𝑅 ↔ [𝑥]𝑅 ≠ ∅) | |
6 | 4, 5 | sylib 207 | . 2 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑥]𝑅 ≠ ∅) |
7 | 1, 2, 6 | ectocld 7701 | 1 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 dom cdm 5038 [cec 7627 / cqs 7628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ec 7631 df-qs 7635 |
This theorem is referenced by: ecelqsdm 7704 0nsr 9779 sylow1lem3 17838 vitalilem5 23187 prtlem400 33173 |
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