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| Mirrors > Home > MPE Home > Th. List > difsymssdifssd | Structured version Visualization version GIF version | ||
| Description: If the symmetric difference is contained in 𝐶, so is one of the differences. (Contributed by AV, 17-Aug-2022.) | 
| Ref | Expression | 
|---|---|
| difsymssdifssd.1 | ⊢ (𝜑 → (𝐴 △ 𝐵) ⊆ 𝐶) | 
| Ref | Expression | 
|---|---|
| difsymssdifssd | ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | difsssymdif 4263 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ (𝐴 △ 𝐵) | |
| 2 | difsymssdifssd.1 | . 2 ⊢ (𝜑 → (𝐴 △ 𝐵) ⊆ 𝐶) | |
| 3 | 1, 2 | sstrid 3995 | 1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∖ cdif 3948 ⊆ wss 3951 △ csymdif 4252 | 
| 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 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-symdif 4253 | 
| This theorem is referenced by: mbfeqalem1 25676 | 
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