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Mirrors > Home > ILE Home > Th. List > tfrlem8 | Unicode version |
Description: Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.) |
Ref | Expression |
---|---|
tfrlem.1 |
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Ref | Expression |
---|---|
tfrlem8 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlem.1 |
. . . . . . . . 9
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2 | 1 | tfrlem3 6216 |
. . . . . . . 8
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3 | 2 | abeq2i 2251 |
. . . . . . 7
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4 | fndm 5230 |
. . . . . . . . . . 11
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5 | 4 | adantr 274 |
. . . . . . . . . 10
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6 | 5 | eleq1d 2209 |
. . . . . . . . 9
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7 | 6 | biimprcd 159 |
. . . . . . . 8
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8 | 7 | rexlimiv 2546 |
. . . . . . 7
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9 | 3, 8 | sylbi 120 |
. . . . . 6
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10 | eleq1a 2212 |
. . . . . 6
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11 | 9, 10 | syl 14 |
. . . . 5
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12 | 11 | rexlimiv 2546 |
. . . 4
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13 | 12 | abssi 3177 |
. . 3
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14 | ssorduni 4411 |
. . 3
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15 | 13, 14 | ax-mp 5 |
. 2
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16 | 1 | recsfval 6220 |
. . . . 5
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17 | 16 | dmeqi 4748 |
. . . 4
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18 | dmuni 4757 |
. . . 4
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19 | vex 2692 |
. . . . . 6
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20 | 19 | dmex 4813 |
. . . . 5
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21 | 20 | dfiun2 3855 |
. . . 4
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22 | 17, 18, 21 | 3eqtri 2165 |
. . 3
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23 | ordeq 4302 |
. . 3
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24 | 22, 23 | ax-mp 5 |
. 2
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25 | 15, 24 | mpbir 145 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-tr 4035 df-iord 4296 df-on 4298 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-iota 5096 df-fun 5133 df-fn 5134 df-fv 5139 df-recs 6210 |
This theorem is referenced by: tfrlemi14d 6238 tfri1dALT 6256 |
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