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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 3221 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | difss 3127 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
3 | 1, 2 | unssi 3176 | 1 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∖ cdif 2997 ∪ cun 2998 ∩ cin 2999 ⊆ wss 3000 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 |
This theorem is referenced by: resasplitss 5203 |
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