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Mirrors > Home > ILE Home > Th. List > dfsn2 | GIF version |
Description: Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
Ref | Expression |
---|---|
dfsn2 | ⊢ {𝐴} = {𝐴, 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3590 | . 2 ⊢ {𝐴, 𝐴} = ({𝐴} ∪ {𝐴}) | |
2 | unidm 3270 | . 2 ⊢ ({𝐴} ∪ {𝐴}) = {𝐴} | |
3 | 1, 2 | eqtr2i 2192 | 1 ⊢ {𝐴} = {𝐴, 𝐴} |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∪ cun 3119 {csn 3583 {cpr 3584 |
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-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-pr 3590 |
This theorem is referenced by: nfsn 3643 tpidm12 3682 tpidm 3685 preqsn 3762 opid 3783 unisn 3812 intsng 3865 opeqsn 4237 relop 4761 funopg 5232 enpr1g 6776 hashprg 10743 bj-snexg 13947 |
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