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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 6080. (Contributed by Jim Kingdon, 10-Aug-2018.) |
Ref | Expression |
---|---|
difsnss | ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 3087 | . 2 ⊢ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = ({𝐵} ∪ (𝐴 ∖ {𝐵})) | |
2 | snssi 3508 | . . 3 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
3 | undifss 3303 | . . 3 ⊢ ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∪ (𝐴 ∖ {𝐵})) ⊆ 𝐴) | |
4 | 2, 3 | sylib 127 | . 2 ⊢ (𝐵 ∈ 𝐴 → ({𝐵} ∪ (𝐴 ∖ {𝐵})) ⊆ 𝐴) |
5 | 1, 4 | syl5eqss 2989 | 1 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 ∖ cdif 2914 ∪ cun 2915 ⊆ wss 2917 {csn 3375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 |
This theorem is referenced by: nndifsnid 6080 fidifsnid 6332 |
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