Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sssnr | GIF version |
Description: Empty set and the singleton itself are subsets of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.) |
Ref | Expression |
---|---|
sssnr | ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3255 | . . 3 ⊢ ∅ ⊆ {𝐵} | |
2 | sseq1 2966 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵})) | |
3 | 1, 2 | mpbiri 157 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵}) |
4 | eqimss 2997 | . 2 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵}) | |
5 | 3, 4 | jaoi 636 | 1 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 629 = wceq 1243 ⊆ wss 2917 ∅c0 3224 {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-in 2924 df-ss 2931 df-nul 3225 |
This theorem is referenced by: pwsnss 3574 |
Copyright terms: Public domain | W3C validator |