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Mirrors > Home > ILE Home > Th. List > undifabs | GIF version |
Description: Absorption of difference by union. (Contributed by NM, 18-Aug-2013.) |
Ref | Expression |
---|---|
undifabs | ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3083 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
2 | difss 3168 | . . 3 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
3 | 1, 2 | unssi 3217 | . 2 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴 |
4 | ssun1 3205 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∪ (𝐴 ∖ 𝐵)) | |
5 | 3, 4 | eqssi 3079 | 1 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1314 ∖ cdif 3034 ∪ cun 3035 |
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 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 |
This theorem is referenced by: exmid1stab 12887 |
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