![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > prsstp12 | GIF version |
Description: A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.) |
Ref | Expression |
---|---|
prsstp12 | ⊢ {𝐴, 𝐵} ⊆ {𝐴, 𝐵, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 3310 | . 2 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | df-tp 3612 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
3 | 1, 2 | sseqtrri 3202 | 1 ⊢ {𝐴, 𝐵} ⊆ {𝐴, 𝐵, 𝐶} |
Colors of variables: wff set class |
Syntax hints: ∪ cun 3139 ⊆ wss 3141 {csn 3604 {cpr 3605 {ctp 3606 |
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-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-tp 3612 |
This theorem is referenced by: prsstp13 3758 prsstp23 3759 sstpr 3769 |
Copyright terms: Public domain | W3C validator |