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Mirrors > Home > MPE Home > Th. List > Mathboxes > n0eldmqseq | Structured version Visualization version GIF version |
Description: The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 3-Nov-2018.) |
Ref | Expression |
---|---|
n0eldmqseq | ⊢ ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0eldmqs 36407 | . 2 ⊢ ¬ ∅ ∈ (dom 𝑅 / 𝑅) | |
2 | eleq2 2822 | . 2 ⊢ ((dom 𝑅 / 𝑅) = 𝐴 → (∅ ∈ (dom 𝑅 / 𝑅) ↔ ∅ ∈ 𝐴)) | |
3 | 1, 2 | mtbii 329 | 1 ⊢ ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2114 ∅c0 4212 dom cdm 5526 / cqs 8322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pr 5297 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-sn 4518 df-pr 4520 df-op 4524 df-br 5032 df-opab 5094 df-xp 5532 df-cnv 5534 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-ec 8325 df-qs 8329 |
This theorem is referenced by: n0el3 36409 |
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