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Mirrors > Home > ILE Home > Th. List > ssprr | GIF version |
Description: The subsets of a pair. (Contributed by Jim Kingdon, 11-Aug-2018.) |
Ref | Expression |
---|---|
ssprr | ⊢ (((A = ∅ ∨ A = {B}) ∨ (A = {𝐶} ∨ A = {B, 𝐶})) → A ⊆ {B, 𝐶}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3249 | . . . 4 ⊢ ∅ ⊆ {B, 𝐶} | |
2 | sseq1 2960 | . . . 4 ⊢ (A = ∅ → (A ⊆ {B, 𝐶} ↔ ∅ ⊆ {B, 𝐶})) | |
3 | 1, 2 | mpbiri 157 | . . 3 ⊢ (A = ∅ → A ⊆ {B, 𝐶}) |
4 | snsspr1 3503 | . . . 4 ⊢ {B} ⊆ {B, 𝐶} | |
5 | sseq1 2960 | . . . 4 ⊢ (A = {B} → (A ⊆ {B, 𝐶} ↔ {B} ⊆ {B, 𝐶})) | |
6 | 4, 5 | mpbiri 157 | . . 3 ⊢ (A = {B} → A ⊆ {B, 𝐶}) |
7 | 3, 6 | jaoi 635 | . 2 ⊢ ((A = ∅ ∨ A = {B}) → A ⊆ {B, 𝐶}) |
8 | snsspr2 3504 | . . . 4 ⊢ {𝐶} ⊆ {B, 𝐶} | |
9 | sseq1 2960 | . . . 4 ⊢ (A = {𝐶} → (A ⊆ {B, 𝐶} ↔ {𝐶} ⊆ {B, 𝐶})) | |
10 | 8, 9 | mpbiri 157 | . . 3 ⊢ (A = {𝐶} → A ⊆ {B, 𝐶}) |
11 | eqimss 2991 | . . 3 ⊢ (A = {B, 𝐶} → A ⊆ {B, 𝐶}) | |
12 | 10, 11 | jaoi 635 | . 2 ⊢ ((A = {𝐶} ∨ A = {B, 𝐶}) → A ⊆ {B, 𝐶}) |
13 | 7, 12 | jaoi 635 | 1 ⊢ (((A = ∅ ∨ A = {B}) ∨ (A = {𝐶} ∨ A = {B, 𝐶})) → A ⊆ {B, 𝐶}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 628 = wceq 1242 ⊆ wss 2911 ∅c0 3218 {csn 3367 {cpr 3368 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pr 3374 |
This theorem is referenced by: sstpr 3519 pwprss 3567 |
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