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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 3985 | . 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 2108 ⊆ wss 3951 |
| 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 1967 ax-7 2007 ax-8 2110 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-clel 2816 df-ss 3968 |
| This theorem is referenced by: 0nelrel0 5745 cantnfp1lem3 9720 fpwwe2lem12 10682 pwfseqlem3 10700 hashbclem 14491 sumrblem 15747 incexclem 15872 prodrblem 15965 fprodntriv 15978 ramub1lem2 17065 mreexmrid 17686 mreexexlem2d 17688 acsfiindd 18598 lbspss 21081 lbsextlem4 21163 lindfrn 21841 fclscmpi 24037 lhop2 26054 lhop 26055 dvcnvrelem1 26056 axlowdimlem17 28973 cyc3co2 33160 ssdifidlprm 33486 erdszelem8 35203 bj-fununsn1 37254 bj-fvsnun2 37257 poimirlem16 37643 osumcllem10N 39967 pexmidlem7N 39978 mapdindp2 41723 mapdindp3 41724 hdmapval3lemN 41839 hdmap11lem1 41843 fourierdlem80 46201 |
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