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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 6337 |
. . . . . . . 8
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3 | 2 | abeq2i 2300 |
. . . . . . 7
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4 | fndm 5334 |
. . . . . . . . . . 11
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5 | 4 | adantr 276 |
. . . . . . . . . 10
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6 | 5 | eleq1d 2258 |
. . . . . . . . 9
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7 | 6 | biimprcd 160 |
. . . . . . . 8
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8 | 7 | rexlimiv 2601 |
. . . . . . 7
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9 | 3, 8 | sylbi 121 |
. . . . . 6
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10 | eleq1a 2261 |
. . . . . 6
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11 | 9, 10 | syl 14 |
. . . . 5
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12 | 11 | rexlimiv 2601 |
. . . 4
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13 | 12 | abssi 3245 |
. . 3
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14 | ssorduni 4504 |
. . 3
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15 | 13, 14 | ax-mp 5 |
. 2
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16 | 1 | recsfval 6341 |
. . . . 5
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17 | 16 | dmeqi 4846 |
. . . 4
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18 | dmuni 4855 |
. . . 4
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19 | vex 2755 |
. . . . . 6
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20 | 19 | dmex 4911 |
. . . . 5
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21 | 20 | dfiun2 3935 |
. . . 4
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22 | 17, 18, 21 | 3eqtri 2214 |
. . 3
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23 | ordeq 4390 |
. . 3
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24 | 22, 23 | ax-mp 5 |
. 2
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25 | 15, 24 | mpbir 146 |
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-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 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 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-ral 2473 df-rex 2474 df-v 2754 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-iun 3903 df-br 4019 df-opab 4080 df-tr 4117 df-iord 4384 df-on 4386 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-iota 5196 df-fun 5237 df-fn 5238 df-fv 5243 df-recs 6331 |
This theorem is referenced by: tfrlemi14d 6359 tfri1dALT 6377 |
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