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Mirrors > Home > MPE Home > Th. List > difsssymdif | Structured version Visualization version GIF version |
Description: The symmetric difference contains one of the differences. (Proposed by BJ, 18-Aug-2022.) (Contributed by AV, 19-Aug-2022.) |
Ref | Expression |
---|---|
difsssymdif | ⊢ (𝐴 ∖ 𝐵) ⊆ (𝐴 △ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 4148 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) | |
2 | df-symdif 4219 | . 2 ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) | |
3 | 1, 2 | sseqtrri 4004 | 1 ⊢ (𝐴 ∖ 𝐵) ⊆ (𝐴 △ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∖ cdif 3933 ∪ cun 3934 ⊆ wss 3936 △ csymdif 4218 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-un 3941 df-in 3943 df-ss 3952 df-symdif 4219 |
This theorem is referenced by: difsymssdifssd 4230 |
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