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Mirrors > Home > ILE Home > Th. List > en1uniel | Unicode version |
Description: A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
Ref | Expression |
---|---|
en1uniel |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relen 6774 |
. . . 4
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2 | 1 | brrelex1i 4690 |
. . 3
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3 | uniexg 4460 |
. . 3
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4 | snidg 3639 |
. . 3
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5 | 2, 3, 4 | 3syl 17 |
. 2
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6 | encv 6776 |
. . . . 5
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7 | 6 | simpld 112 |
. . . 4
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8 | en1bg 6830 |
. . . 4
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9 | 7, 8 | syl 14 |
. . 3
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10 | 9 | ibi 176 |
. 2
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11 | 5, 10 | eleqtrrd 2269 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-in1 615 ax-in2 616 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-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-id 4314 df-suc 4392 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-1o 6445 df-en 6771 |
This theorem is referenced by: en2eleq 7229 en2other2 7230 |
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