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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 35917 | . 2 ⊢ ¬ ∅ ∈ (dom 𝑅 / 𝑅) | |
2 | eleq2 2900 | . 2 ⊢ ((dom 𝑅 / 𝑅) = 𝐴 → (∅ ∈ (dom 𝑅 / 𝑅) ↔ ∅ ∈ 𝐴)) | |
3 | 1, 2 | mtbii 328 | 1 ⊢ ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2113 ∅c0 4284 dom cdm 5548 / cqs 8281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ec 8284 df-qs 8288 |
This theorem is referenced by: n0el3 35919 |
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