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Mirrors > Home > ILE Home > Th. List > uzn0 | Unicode version |
Description: The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.) |
Ref | Expression |
---|---|
uzn0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzf 9520 |
. . 3
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2 | ffn 5361 |
. . 3
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3 | fvelrnb 5559 |
. . 3
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4 | 1, 2, 3 | mp2b 8 |
. 2
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5 | uzid 9531 |
. . . . 5
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6 | ne0i 3429 |
. . . . 5
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7 | 5, 6 | syl 14 |
. . . 4
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8 | neeq1 2360 |
. . . 4
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9 | 7, 8 | syl5ibcom 155 |
. . 3
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10 | 9 | rexlimiv 2588 |
. 2
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11 | 4, 10 | sylbi 121 |
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-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-pre-ltirr 7914 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 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-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 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-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fv 5220 df-ov 5872 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-neg 8121 df-z 9243 df-uz 9518 |
This theorem is referenced by: (None) |
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