![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 6531. (Contributed by Jim Kingdon, 10-Aug-2018.) |
Ref | Expression |
---|---|
difsnss | ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 3294 | . 2 ⊢ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = ({𝐵} ∪ (𝐴 ∖ {𝐵})) | |
2 | snssi 3751 | . . 3 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
3 | undifss 3518 | . . 3 ⊢ ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∪ (𝐴 ∖ {𝐵})) ⊆ 𝐴) | |
4 | 2, 3 | sylib 122 | . 2 ⊢ (𝐵 ∈ 𝐴 → ({𝐵} ∪ (𝐴 ∖ {𝐵})) ⊆ 𝐴) |
5 | 1, 4 | eqsstrid 3216 | 1 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 ∖ cdif 3141 ∪ cun 3142 ⊆ wss 3144 {csn 3607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 |
This theorem is referenced by: fnsnsplitss 5735 dcdifsnid 6528 |
Copyright terms: Public domain | W3C validator |