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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 3982 | . 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)) |
| 3 | 2 | con3d 152 | 1 ⊢ (𝜑 → (¬ 𝐶 ∈ 𝐵 → ¬ 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: ssneldd 3986 kmlem2 10192 hashbclem 14491 prodss 15983 coprmproddvdslem 16699 mrissmrid 17684 mpfrcl 22109 onsuct0 36442 ftc1anc 37708 dvhdimlem 41446 dvh3dim2 41450 dvh3dim3N 41451 mapdh9a 41791 hdmapval0 41835 hdmap11lem2 41844 iundjiunlem 46474 elbigolo1 48478 |
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