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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 4269 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ (𝐴 △ 𝐵) | |
2 | difsymssdifssd.1 | . 2 ⊢ (𝜑 → (𝐴 △ 𝐵) ⊆ 𝐶) | |
3 | 1, 2 | sstrid 4007 | 1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∖ cdif 3960 ⊆ wss 3963 △ csymdif 4258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-ss 3980 df-symdif 4259 |
This theorem is referenced by: mbfeqalem1 25690 |
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