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Mirrors > Home > ILE Home > Th. List > diffisn | Unicode version |
Description: Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.) |
Ref | Expression |
---|---|
diffisn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6788 |
. . . 4
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2 | 1 | biimpi 120 |
. . 3
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3 | 2 | adantr 276 |
. 2
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4 | elex2 2768 |
. . . . . . . . 9
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5 | 4 | adantl 277 |
. . . . . . . 8
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6 | fin0 6914 |
. . . . . . . . 9
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7 | 6 | adantr 276 |
. . . . . . . 8
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8 | 5, 7 | mpbird 167 |
. . . . . . 7
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9 | 8 | adantr 276 |
. . . . . 6
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10 | 9 | neneqd 2381 |
. . . . 5
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11 | simplrr 536 |
. . . . . . 7
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12 | en0 6822 |
. . . . . . . . 9
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13 | 12 | biimpri 133 |
. . . . . . . 8
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14 | 13 | adantl 277 |
. . . . . . 7
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15 | entr 6811 |
. . . . . . 7
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16 | 11, 14, 15 | syl2anc 411 |
. . . . . 6
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17 | en0 6822 |
. . . . . 6
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18 | 16, 17 | sylib 122 |
. . . . 5
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19 | 10, 18 | mtand 666 |
. . . 4
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20 | nn0suc 4621 |
. . . . . 6
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21 | 20 | orcomd 730 |
. . . . 5
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22 | 21 | ad2antrl 490 |
. . . 4
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23 | 19, 22 | ecased 1360 |
. . 3
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24 | nnfi 6901 |
. . . . 5
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25 | 24 | ad2antrl 490 |
. . . 4
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26 | simprl 529 |
. . . . 5
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27 | simplrr 536 |
. . . . . 6
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28 | breq2 4022 |
. . . . . . 7
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29 | 28 | ad2antll 491 |
. . . . . 6
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30 | 27, 29 | mpbid 147 |
. . . . 5
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31 | simpllr 534 |
. . . . 5
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32 | dif1en 6908 |
. . . . 5
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33 | 26, 30, 31, 32 | syl3anc 1249 |
. . . 4
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34 | enfii 6903 |
. . . 4
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35 | 25, 33, 34 | syl2anc 411 |
. . 3
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36 | 23, 35 | rexlimddv 2612 |
. 2
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37 | 3, 36 | rexlimddv 2612 |
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-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-dc 836 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-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 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-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-er 6560 df-en 6768 df-fin 6770 |
This theorem is referenced by: diffifi 6923 zfz1isolemsplit 10853 zfz1isolem1 10855 fsumdifsnconst 11498 fprodeq0g 11681 |
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