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Mirrors > Home > MPE Home > Th. List > ssneldd | Structured version Visualization version GIF version |
Description: If an element is not in a class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
ssneld.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
ssneldd.2 | ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) |
Ref | Expression |
---|---|
ssneldd | ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssneldd.2 | . 2 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) | |
2 | ssneld.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
3 | 2 | ssneld 3570 | . 2 ⊢ (𝜑 → (¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴)) |
4 | 1, 3 | mpd 15 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1977 ⊆ wss 3540 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-in 3547 df-ss 3554 |
This theorem is referenced by: cantnfp1lem3 8460 fpwwe2lem13 9343 pwfseqlem3 9361 hashbclem 13093 sumrblem 14289 incexclem 14407 prodrblem 14498 fprodntriv 14511 ramub1lem2 15569 mreexmrid 16126 mreexexlem2d 16128 acsfiindd 17000 lbspss 18903 lbsextlem4 18982 lindfrn 19979 fclscmpi 21643 lhop2 23582 lhop 23583 dvcnvrelem1 23584 axlowdimlem17 25638 erdszelem8 30434 osumcllem10N 34269 pexmidlem7N 34280 mapdindp2 36028 mapdindp3 36029 hdmapval3lemN 36147 hdmap11lem1 36151 fourierdlem80 39079 |
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