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Mirrors > Home > ILE Home > Th. List > undif1ss | GIF version |
Description: Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.) |
Ref | Expression |
---|---|
undif1ss | ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) ⊆ (𝐴 ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 3110 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
2 | unss1 3153 | . 2 ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐴 → ((𝐴 ∖ 𝐵) ∪ 𝐵) ⊆ (𝐴 ∪ 𝐵)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) ⊆ (𝐴 ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∖ cdif 2981 ∪ cun 2982 ⊆ wss 2984 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2614 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 |
This theorem is referenced by: undif2ss 3340 pwundifss 4076 |
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