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Mirrors > Home > MPE Home > Th. List > ssdif2d | Structured version Visualization version GIF version |
Description: If 𝐴 is contained in 𝐵 and 𝐶 is contained in 𝐷, then (𝐴 ∖ 𝐷) is contained in (𝐵 ∖ 𝐶). Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
ssdifd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
ssdif2d.2 | ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Ref | Expression |
---|---|
ssdif2d | ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐵 ∖ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdif2d.2 | . . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐷) | |
2 | 1 | sscond 4088 | . 2 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐴 ∖ 𝐶)) |
3 | ssdifd.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
4 | 3 | ssdifd 4087 | . 2 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) |
5 | 2, 4 | sstrd 3942 | 1 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐵 ∖ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∖ cdif 3895 ⊆ wss 3898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3443 df-dif 3901 df-in 3905 df-ss 3915 |
This theorem is referenced by: mblfinlem3 35929 mblfinlem4 35930 |
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