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Mirrors > Home > MPE Home > Th. List > ssneld | Structured version Visualization version GIF version |
Description: If a class is not in another 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) |
Ref | Expression |
---|---|
ssneld | ⊢ (𝜑 → (¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssneld.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | 1 | sseld 3914 | . 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)) |
3 | 2 | con3d 155 | 1 ⊢ (𝜑 → (¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2111 ⊆ wss 3881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 |
This theorem is referenced by: ssneldd 3918 kmlem2 9562 hashbclem 13806 prodss 15293 coprmproddvdslem 15996 mrissmrid 16904 mpfrcl 20757 onsuct0 33902 ftc1anc 35138 dvhdimlem 38740 dvh3dim2 38744 dvh3dim3N 38745 mapdh9a 39085 hdmapval0 39129 hdmap11lem2 39138 iundjiunlem 43098 elbigolo1 44971 |
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