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Mirrors > Home > ILE Home > Th. List > difid | GIF version |
Description: The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.) |
Ref | Expression |
---|---|
difid | ⊢ (𝐴 ∖ 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3087 | . 2 ⊢ 𝐴 ⊆ 𝐴 | |
2 | ssdif0im 3397 | . 2 ⊢ (𝐴 ⊆ 𝐴 → (𝐴 ∖ 𝐴) = ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∖ 𝐴) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ∖ cdif 3038 ⊆ wss 3041 ∅c0 3333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-dif 3043 df-in 3047 df-ss 3054 df-nul 3334 |
This theorem is referenced by: dif0 3403 difun2 3412 diftpsn3 3631 2oconcl 6304 ismkvnex 6997 topcld 12205 |
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