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Mirrors > Home > MPE Home > Th. List > dfsn2 | Structured version Visualization version 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 4634 | . 2 ⊢ {𝐴, 𝐴} = ({𝐴} ∪ {𝐴}) | |
2 | unidm 4167 | . 2 ⊢ ({𝐴} ∪ {𝐴}) = {𝐴} | |
3 | 1, 2 | eqtr2i 2764 | 1 ⊢ {𝐴} = {𝐴, 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∪ cun 3961 {csn 4631 {cpr 4633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-pr 4634 |
This theorem is referenced by: nfsn 4712 disjprsn 4719 tpidm12 4760 tpidm 4763 ifpprsnss 4769 preqsnd 4864 elpreqprlem 4871 opidg 4897 unisng 4930 intsng 4988 vsnex 5440 opeqsng 5513 propeqop 5517 relop 5864 funopg 6602 f1oprswap 6893 fnprb 7228 enpr1g 9062 prfi 9361 supsn 9510 infsn 9543 pr2ne 10042 prdom2 10044 wuntp 10749 wunsn 10754 grusn 10842 prunioo 13518 hashprg 14431 hashfun 14473 hashle2pr 14513 lcmfsn 16669 lubsn 18540 indislem 23023 hmphindis 23821 wilthlem2 27127 negs1s 28074 upgrex 29124 umgrnloop0 29141 edglnl 29175 usgrnloop0ALT 29237 uspgr1v1eop 29281 1loopgruspgr 29533 1egrvtxdg0 29544 umgr2v2eedg 29557 umgr2v2e 29558 ifpsnprss 29656 upgriswlk 29674 clwwlkn1 30070 upgr1wlkdlem1 30174 1to2vfriswmgr 30308 esumpr2 34048 dvh2dim 41428 wopprc 43019 clsk1indlem4 44034 sge0prle 46357 meadjun 46418 elsprel 47400 sclnbgrelself 47772 upgrwlkupwlk 47984 |
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