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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 3567 | . 2 ⊢ {𝐴, 𝐴} = ({𝐴} ∪ {𝐴}) | |
2 | unidm 3250 | . 2 ⊢ ({𝐴} ∪ {𝐴}) = {𝐴} | |
3 | 1, 2 | eqtr2i 2179 | 1 ⊢ {𝐴} = {𝐴, 𝐴} |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∪ cun 3100 {csn 3560 {cpr 3561 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-un 3106 df-pr 3567 |
This theorem is referenced by: nfsn 3619 tpidm12 3658 tpidm 3661 preqsn 3738 opid 3759 unisn 3788 intsng 3841 opeqsn 4212 relop 4736 funopg 5204 enpr1g 6743 hashprg 10682 bj-snexg 13498 |
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