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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 3614 | . 2 ⊢ {𝐴, 𝐴} = ({𝐴} ∪ {𝐴}) | |
2 | unidm 3293 | . 2 ⊢ ({𝐴} ∪ {𝐴}) = {𝐴} | |
3 | 1, 2 | eqtr2i 2211 | 1 ⊢ {𝐴} = {𝐴, 𝐴} |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∪ cun 3142 {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-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-un 3148 df-pr 3614 |
This theorem is referenced by: nfsn 3667 tpidm12 3706 tpidm 3709 preqsn 3790 opid 3811 unisn 3840 intsng 3893 opeqsn 4270 relop 4795 funopg 5269 enpr1g 6824 hashprg 10820 bj-snexg 15122 |
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