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Mirrors > Home > ILE Home > Th. List > nnsseleq | Unicode version |
Description: For natural numbers, inclusion is equivalent to membership or equality. (Contributed by Jim Kingdon, 16-Sep-2021.) |
Ref | Expression |
---|---|
nnsseleq |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nntri1 6400 |
. . 3
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2 | nntri3or 6397 |
. . . . . 6
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3 | df-3or 964 |
. . . . . 6
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4 | 2, 3 | sylib 121 |
. . . . 5
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5 | 4 | orcomd 719 |
. . . 4
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6 | 5 | ord 714 |
. . 3
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7 | 1, 6 | sylbid 149 |
. 2
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8 | nnord 4533 |
. . . . 5
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9 | 8 | adantl 275 |
. . . 4
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10 | ordelss 4309 |
. . . . 5
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11 | 10 | ex 114 |
. . . 4
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12 | 9, 11 | syl 14 |
. . 3
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13 | eqimss 3156 |
. . . 4
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14 | 13 | a1i 9 |
. . 3
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15 | 12, 14 | jaod 707 |
. 2
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16 | 7, 15 | impbid 128 |
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-in1 604 ax-in2 605 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-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-uni 3745 df-int 3780 df-tr 4035 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 |
This theorem is referenced by: nnsssuc 6406 frec2uzled 10233 |
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