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Mirrors > Home > ILE Home > Th. List > inundifss | GIF version |
Description: The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.) |
Ref | Expression |
---|---|
inundifss | ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 3367 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | difss 3273 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
3 | 1, 2 | unssi 3322 | 1 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∖ cdif 3138 ∪ cun 3139 ∩ cin 3140 ⊆ wss 3141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 |
This theorem is referenced by: resasplitss 5407 |
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