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Mirrors > Home > ILE Home > Th. List > ordiso | Unicode version |
Description: Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
ordiso |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resiexg 4757 |
. . . . 5
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2 | isoid 5589 |
. . . . 5
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3 | isoeq1 5580 |
. . . . . 6
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4 | 3 | spcegv 2707 |
. . . . 5
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5 | 1, 2, 4 | mpisyl 1380 |
. . . 4
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6 | 5 | adantr 270 |
. . 3
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7 | isoeq5 5584 |
. . . 4
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8 | 7 | exbidv 1753 |
. . 3
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9 | 6, 8 | syl5ibcom 153 |
. 2
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10 | eloni 4202 |
. . . 4
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11 | eloni 4202 |
. . . 4
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12 | ordiso2 6728 |
. . . . . 6
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13 | 12 | 3coml 1150 |
. . . . 5
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14 | 13 | 3expia 1145 |
. . . 4
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15 | 10, 11, 14 | syl2an 283 |
. . 3
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16 | 15 | exlimdv 1747 |
. 2
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17 | 9, 16 | impbid 127 |
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 ax-setind 4353 |
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-rab 2368 df-v 2621 df-sbc 2841 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-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-eprel 4116 df-id 4120 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-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-isom 5024 |
This theorem is referenced by: (None) |
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