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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 3370 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | difss 3276 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
3 | 1, 2 | unssi 3325 | 1 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∖ cdif 3141 ∪ cun 3142 ∩ cin 3143 ⊆ wss 3144 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 |
This theorem is referenced by: resasplitss 5410 |
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