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Mirrors > Home > ILE Home > Th. List > nndifsnid | Unicode version |
Description: If we remove a single element from a natural number then put it back in, we end up with the original natural number. This strengthens difsnss 3674 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.) |
Ref | Expression |
---|---|
nndifsnid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn 4527 |
. . . . . 6
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2 | 1 | expcom 115 |
. . . . 5
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3 | elnn 4527 |
. . . . . 6
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4 | 3 | expcom 115 |
. . . . 5
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5 | 2, 4 | anim12d 333 |
. . . 4
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6 | nndceq 6403 |
. . . 4
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7 | 5, 6 | syl6 33 |
. . 3
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8 | 7 | ralrimivv 2516 |
. 2
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9 | dcdifsnid 6408 |
. 2
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10 | 8, 9 | sylan 281 |
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-dc 821 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: phplem2 6755 |
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