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Mirrors > Home > ILE Home > Th. List > nnwetri | Unicode version |
Description: A natural number is
well-ordered by ![]() ![]() |
Ref | Expression |
---|---|
nnwetri |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnord 4608 |
. . 3
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2 | ordwe 4572 |
. . 3
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3 | 1, 2 | syl 14 |
. 2
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4 | simprl 529 |
. . . . 5
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5 | simpl 109 |
. . . . 5
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6 | elnn 4602 |
. . . . 5
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7 | 4, 5, 6 | syl2anc 411 |
. . . 4
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8 | simprr 531 |
. . . . 5
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9 | elnn 4602 |
. . . . 5
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10 | 8, 5, 9 | syl2anc 411 |
. . . 4
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11 | nntri3or 6488 |
. . . . 5
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12 | epel 4289 |
. . . . . 6
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13 | biid 171 |
. . . . . 6
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14 | epel 4289 |
. . . . . 6
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15 | 12, 13, 14 | 3orbi123i 1189 |
. . . . 5
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16 | 11, 15 | sylibr 134 |
. . . 4
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17 | 7, 10, 16 | syl2anc 411 |
. . 3
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18 | 17 | ralrimivva 2559 |
. 2
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19 | 3, 18 | jca 306 |
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-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 |
This theorem depends on definitions: df-bi 117 df-3or 979 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-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-int 3843 df-br 4001 df-opab 4062 df-tr 4099 df-eprel 4286 df-frfor 4328 df-frind 4329 df-wetr 4331 df-iord 4363 df-on 4365 df-suc 4368 df-iom 4587 |
This theorem is referenced by: (None) |
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