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Mirrors > Home > ILE Home > Th. List > sbthlemi6 | Unicode version |
Description: Lemma for isbth 6960. (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 527 |
. . 3
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2 | simprll 537 |
. . 3
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3 | simprlr 538 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | simprr 531 |
. . 3
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5 | rnun 5033 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | sbthlem.3 |
. . . . . 6
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7 | 6 | rneqi 4851 |
. . . . 5
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8 | df-ima 4636 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 8 | uneq1i 3285 |
. . . . 5
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10 | 5, 7, 9 | 3eqtr4i 2208 |
. . . 4
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11 | sbthlem.1 |
. . . . . . 7
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12 | sbthlem.2 |
. . . . . . 7
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13 | 11, 12 | sbthlemi4 6953 |
. . . . . 6
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14 | df-ima 4636 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 13, 14 | eqtr3di 2225 |
. . . . 5
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16 | 15 | uneq2d 3289 |
. . . 4
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17 | 10, 16 | eqtr4id 2229 |
. . 3
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18 | 1, 2, 3, 4, 17 | syl121anc 1243 |
. 2
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19 | imassrn 4977 |
. . . . . . 7
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20 | sstr2 3162 |
. . . . . . 7
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21 | 19, 20 | ax-mp 5 |
. . . . . 6
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22 | 21 | adantl 277 |
. . . . 5
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23 | undifdcss 6916 |
. . . . . . 7
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24 | exmidexmid 4193 |
. . . . . . . . 9
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25 | 24 | ralrimivw 2551 |
. . . . . . . 8
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26 | 25 | biantrud 304 |
. . . . . . 7
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27 | 23, 26 | bitr4id 199 |
. . . . . 6
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28 | 27 | adantr 276 |
. . . . 5
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29 | 22, 28 | mpbird 167 |
. . . 4
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30 | 29 | eqcomd 2183 |
. . 3
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31 | 30 | adantr 276 |
. 2
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32 | 18, 31 | eqtrd 2210 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-exmid 4192 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-fun 5214 |
This theorem is referenced by: sbthlemi9 6958 |
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