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Mirrors > Home > ILE Home > Th. List > difsnss | GIF version |
Description: If we remove a single element from a class then put it back in, we end up with a subset of the original class. If equality is decidable, we can replace subset with equality as seen in nndifsnid 6306. (Contributed by Jim Kingdon, 10-Aug-2018.) |
Ref | Expression |
---|---|
difsnss | ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 3159 | . 2 ⊢ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = ({𝐵} ∪ (𝐴 ∖ {𝐵})) | |
2 | snssi 3603 | . . 3 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
3 | undifss 3382 | . . 3 ⊢ ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∪ (𝐴 ∖ {𝐵})) ⊆ 𝐴) | |
4 | 2, 3 | sylib 121 | . 2 ⊢ (𝐵 ∈ 𝐴 → ({𝐵} ∪ (𝐴 ∖ {𝐵})) ⊆ 𝐴) |
5 | 1, 4 | syl5eqss 3085 | 1 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1445 ∖ cdif 3010 ∪ cun 3011 ⊆ wss 3013 {csn 3466 |
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 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-sn 3472 |
This theorem is referenced by: fnsnsplitss 5535 dcdifsnid 6303 |
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