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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 3476 | . . . 4 ⊢ ∅ ⊆ {𝐵, 𝐶} | |
2 | sseq1 3193 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵, 𝐶} ↔ ∅ ⊆ {𝐵, 𝐶})) | |
3 | 1, 2 | mpbiri 168 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵, 𝐶}) |
4 | snsspr1 3755 | . . . 4 ⊢ {𝐵} ⊆ {𝐵, 𝐶} | |
5 | sseq1 3193 | . . . 4 ⊢ (𝐴 = {𝐵} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐵} ⊆ {𝐵, 𝐶})) | |
6 | 4, 5 | mpbiri 168 | . . 3 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵, 𝐶}) |
7 | 3, 6 | jaoi 717 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵, 𝐶}) |
8 | snsspr2 3756 | . . . 4 ⊢ {𝐶} ⊆ {𝐵, 𝐶} | |
9 | sseq1 3193 | . . . 4 ⊢ (𝐴 = {𝐶} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐶} ⊆ {𝐵, 𝐶})) | |
10 | 8, 9 | mpbiri 168 | . . 3 ⊢ (𝐴 = {𝐶} → 𝐴 ⊆ {𝐵, 𝐶}) |
11 | eqimss 3224 | . . 3 ⊢ (𝐴 = {𝐵, 𝐶} → 𝐴 ⊆ {𝐵, 𝐶}) | |
12 | 10, 11 | jaoi 717 | . 2 ⊢ ((𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶}) → 𝐴 ⊆ {𝐵, 𝐶}) |
13 | 7, 12 | jaoi 717 | 1 ⊢ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 709 = wceq 1364 ⊆ wss 3144 ∅c0 3437 {csn 3607 {cpr 3608 |
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-nul 3438 df-pr 3614 |
This theorem is referenced by: sstpr 3772 pwprss 3820 |
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