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Mirrors > Home > ILE Home > Th. List > sbthlemi6 | Unicode version |
Description: Lemma for isbth 6807. (Contributed by NM, 27-Mar-1998.) |
Ref | Expression |
---|---|
sbthlem.1 |
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sbthlem.2 |
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sbthlem.3 |
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Ref | Expression |
---|---|
sbthlemi6 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 501 |
. . 3
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2 | simprll 509 |
. . 3
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3 | simprlr 510 |
. . 3
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4 | simprr 504 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | df-ima 4512 |
. . . . . 6
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6 | sbthlem.1 |
. . . . . . 7
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7 | sbthlem.2 |
. . . . . . 7
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8 | 6, 7 | sbthlemi4 6800 |
. . . . . 6
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9 | 5, 8 | syl5reqr 2162 |
. . . . 5
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10 | 9 | uneq2d 3196 |
. . . 4
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11 | rnun 4905 |
. . . . 5
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12 | sbthlem.3 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 12 | rneqi 4727 |
. . . . 5
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14 | df-ima 4512 |
. . . . . 6
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15 | 14 | uneq1i 3192 |
. . . . 5
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16 | 11, 13, 15 | 3eqtr4i 2145 |
. . . 4
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17 | 10, 16 | syl6reqr 2166 |
. . 3
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18 | 1, 2, 3, 4, 17 | syl121anc 1204 |
. 2
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19 | imassrn 4850 |
. . . . . . 7
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20 | sstr2 3070 |
. . . . . . 7
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21 | 19, 20 | ax-mp 7 |
. . . . . 6
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22 | 21 | adantl 273 |
. . . . 5
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23 | exmidexmid 4080 |
. . . . . . . . 9
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24 | 23 | ralrimivw 2480 |
. . . . . . . 8
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25 | 24 | biantrud 300 |
. . . . . . 7
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26 | undifdcss 6764 |
. . . . . . 7
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27 | 25, 26 | syl6rbbr 198 |
. . . . . 6
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28 | 27 | adantr 272 |
. . . . 5
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29 | 22, 28 | mpbird 166 |
. . . 4
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30 | 29 | eqcomd 2120 |
. . 3
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31 | 30 | adantr 272 |
. 2
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32 | 18, 31 | eqtrd 2147 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-nul 4014 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-stab 799 df-dc 803 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-exmid 4079 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-fun 5083 |
This theorem is referenced by: sbthlemi9 6805 |
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