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Mirrors > Home > ILE Home > Th. List > fzo0to42pr | Unicode version |
Description: A half-open integer range from 0 to 4 is a union of two unordered pairs. (Contributed by Alexander van der Vekens, 17-Nov-2017.) |
Ref | Expression |
---|---|
fzo0to42pr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 9207 |
. . . 4
![]() ![]() ![]() ![]() | |
2 | 4nn0 9209 |
. . . 4
![]() ![]() ![]() ![]() | |
3 | 2re 9003 |
. . . . 5
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4 | 4re 9010 |
. . . . 5
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5 | 2lt4 9106 |
. . . . 5
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6 | 3, 4, 5 | ltleii 8074 |
. . . 4
![]() ![]() ![]() ![]() |
7 | elfz2nn0 10126 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 1, 2, 6, 7 | mpbir3an 1180 |
. . 3
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9 | fzosplit 10191 |
. . 3
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10 | 8, 9 | ax-mp 5 |
. 2
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11 | fzo0to2pr 10232 |
. . 3
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12 | 4z 9297 |
. . . . 5
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13 | fzoval 10162 |
. . . . 5
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14 | 12, 13 | ax-mp 5 |
. . . 4
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15 | 4cn 9011 |
. . . . . . . 8
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16 | ax-1cn 7918 |
. . . . . . . 8
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17 | 3cn 9008 |
. . . . . . . 8
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18 | df-4 8994 |
. . . . . . . . . 10
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19 | 17, 16 | addcomi 8115 |
. . . . . . . . . 10
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20 | 18, 19 | eqtri 2208 |
. . . . . . . . 9
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21 | 20 | eqcomi 2191 |
. . . . . . . 8
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22 | 15, 16, 17, 21 | subaddrii 8260 |
. . . . . . 7
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23 | df-3 8993 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 22, 23 | eqtri 2208 |
. . . . . 6
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25 | 24 | oveq2i 5899 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 2z 9295 |
. . . . . 6
![]() ![]() ![]() ![]() | |
27 | fzpr 10091 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 26, 27 | ax-mp 5 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 25, 28 | eqtri 2208 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 23 | eqcomi 2191 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 30 | preq2i 3685 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 14, 29, 31 | 3eqtri 2212 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 11, 32 | uneq12i 3299 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 10, 33 | eqtri 2208 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-distr 7929 ax-i2m1 7930 ax-0lt1 7931 ax-0id 7933 ax-rnegex 7934 ax-cnre 7936 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-lttrn 7939 ax-pre-apti 7940 ax-pre-ltadd 7941 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-sub 8144 df-neg 8145 df-inn 8934 df-2 8992 df-3 8993 df-4 8994 df-n0 9191 df-z 9268 df-uz 9543 df-fz 10023 df-fzo 10157 |
This theorem is referenced by: (None) |
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