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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 3321 | . . . 4 ⊢ ∅ ⊆ {𝐵, 𝐶} | |
2 | sseq1 3047 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵, 𝐶} ↔ ∅ ⊆ {𝐵, 𝐶})) | |
3 | 1, 2 | mpbiri 166 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵, 𝐶}) |
4 | snsspr1 3585 | . . . 4 ⊢ {𝐵} ⊆ {𝐵, 𝐶} | |
5 | sseq1 3047 | . . . 4 ⊢ (𝐴 = {𝐵} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐵} ⊆ {𝐵, 𝐶})) | |
6 | 4, 5 | mpbiri 166 | . . 3 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵, 𝐶}) |
7 | 3, 6 | jaoi 671 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵, 𝐶}) |
8 | snsspr2 3586 | . . . 4 ⊢ {𝐶} ⊆ {𝐵, 𝐶} | |
9 | sseq1 3047 | . . . 4 ⊢ (𝐴 = {𝐶} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐶} ⊆ {𝐵, 𝐶})) | |
10 | 8, 9 | mpbiri 166 | . . 3 ⊢ (𝐴 = {𝐶} → 𝐴 ⊆ {𝐵, 𝐶}) |
11 | eqimss 3078 | . . 3 ⊢ (𝐴 = {𝐵, 𝐶} → 𝐴 ⊆ {𝐵, 𝐶}) | |
12 | 10, 11 | jaoi 671 | . 2 ⊢ ((𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶}) → 𝐴 ⊆ {𝐵, 𝐶}) |
13 | 7, 12 | jaoi 671 | 1 ⊢ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 664 = wceq 1289 ⊆ wss 2999 ∅c0 3286 {csn 3446 {cpr 3447 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pr 3453 |
This theorem is referenced by: sstpr 3601 pwprss 3649 |
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