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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 3406 | . . . 4 ⊢ ∅ ⊆ {𝐵, 𝐶} | |
2 | sseq1 3125 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵, 𝐶} ↔ ∅ ⊆ {𝐵, 𝐶})) | |
3 | 1, 2 | mpbiri 167 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵, 𝐶}) |
4 | snsspr1 3676 | . . . 4 ⊢ {𝐵} ⊆ {𝐵, 𝐶} | |
5 | sseq1 3125 | . . . 4 ⊢ (𝐴 = {𝐵} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐵} ⊆ {𝐵, 𝐶})) | |
6 | 4, 5 | mpbiri 167 | . . 3 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵, 𝐶}) |
7 | 3, 6 | jaoi 706 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵, 𝐶}) |
8 | snsspr2 3677 | . . . 4 ⊢ {𝐶} ⊆ {𝐵, 𝐶} | |
9 | sseq1 3125 | . . . 4 ⊢ (𝐴 = {𝐶} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐶} ⊆ {𝐵, 𝐶})) | |
10 | 8, 9 | mpbiri 167 | . . 3 ⊢ (𝐴 = {𝐶} → 𝐴 ⊆ {𝐵, 𝐶}) |
11 | eqimss 3156 | . . 3 ⊢ (𝐴 = {𝐵, 𝐶} → 𝐴 ⊆ {𝐵, 𝐶}) | |
12 | 10, 11 | jaoi 706 | . 2 ⊢ ((𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶}) → 𝐴 ⊆ {𝐵, 𝐶}) |
13 | 7, 12 | jaoi 706 | 1 ⊢ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 698 = wceq 1332 ⊆ wss 3076 ∅c0 3368 {csn 3532 {cpr 3533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pr 3539 |
This theorem is referenced by: sstpr 3692 pwprss 3740 |
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