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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 6076 |
. . . . . . . 8
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3 | 2 | abeq2i 2198 |
. . . . . . 7
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4 | fndm 5113 |
. . . . . . . . . . 11
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5 | 4 | adantr 270 |
. . . . . . . . . 10
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6 | 5 | eleq1d 2156 |
. . . . . . . . 9
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7 | 6 | biimprcd 158 |
. . . . . . . 8
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8 | 7 | rexlimiv 2483 |
. . . . . . 7
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9 | 3, 8 | sylbi 119 |
. . . . . 6
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10 | eleq1a 2159 |
. . . . . 6
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11 | 9, 10 | syl 14 |
. . . . 5
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12 | 11 | rexlimiv 2483 |
. . . 4
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13 | 12 | abssi 3096 |
. . 3
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14 | ssorduni 4304 |
. . 3
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15 | 13, 14 | ax-mp 7 |
. 2
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16 | 1 | recsfval 6080 |
. . . . 5
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17 | 16 | dmeqi 4637 |
. . . 4
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18 | dmuni 4646 |
. . . 4
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19 | vex 2622 |
. . . . . 6
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20 | 19 | dmex 4699 |
. . . . 5
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21 | 20 | dfiun2 3764 |
. . . 4
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22 | 17, 18, 21 | 3eqtri 2112 |
. . 3
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23 | ordeq 4199 |
. . 3
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24 | 22, 23 | ax-mp 7 |
. 2
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25 | 15, 24 | mpbir 144 |
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 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-iun 3732 df-br 3846 df-opab 3900 df-tr 3937 df-iord 4193 df-on 4195 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-iota 4980 df-fun 5017 df-fn 5018 df-fv 5023 df-recs 6070 |
This theorem is referenced by: tfrlemi14d 6098 tfri1dALT 6116 |
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