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Mirrors > Home > ILE Home > Th. List > strle1g | Unicode version |
Description: Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.) |
Ref | Expression |
---|---|
strle1.i |
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strle1.a |
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Ref | Expression |
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strle1g |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strle1.i |
. . . 4
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2 | 1 | nnrei 8957 |
. . . . 5
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3 | 2 | leidi 8471 |
. . . 4
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4 | 1, 1, 3 | 3pm3.2i 1177 |
. . 3
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5 | 4 | a1i 9 |
. 2
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6 | difss 3276 |
. . 3
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7 | strle1.a |
. . . . 5
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8 | 7, 1 | eqeltri 2262 |
. . . 4
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9 | funsng 5281 |
. . . 4
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10 | 8, 9 | mpan 424 |
. . 3
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11 | funss 5254 |
. . 3
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12 | 6, 10, 11 | mpsyl 65 |
. 2
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13 | opexg 4246 |
. . . 4
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14 | 8, 13 | mpan 424 |
. . 3
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15 | snexg 4202 |
. . 3
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16 | 14, 15 | syl 14 |
. 2
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17 | dmsnopg 5118 |
. . . 4
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18 | 7 | sneqi 3619 |
. . . . 5
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19 | 1 | nnzi 9303 |
. . . . . 6
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20 | fzsn 10095 |
. . . . . 6
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21 | 19, 20 | ax-mp 5 |
. . . . 5
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22 | 18, 21 | eqtr4i 2213 |
. . . 4
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23 | 17, 22 | eqtrdi 2238 |
. . 3
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24 | eqimss 3224 |
. . 3
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25 | 23, 24 | syl 14 |
. 2
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26 | isstructr 12526 |
. 2
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27 | 5, 12, 16, 25, 26 | syl13anc 1251 |
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 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-addcom 7940 ax-addass 7942 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-0id 7948 ax-rnegex 7949 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-apti 7955 ax-pre-ltadd 7956 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 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-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-inn 8949 df-z 9283 df-uz 9558 df-fz 10038 df-struct 12513 |
This theorem is referenced by: strle2g 12616 strle3g 12617 1strstrg 12625 srngstrd 12654 lmodstrd 12672 |
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