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Mirrors > Home > MPE Home > Th. List > Mathboxes > relexp0eq | Structured version Visualization version Unicode version |
Description: The zeroth power of relationships is the same if and only if the union of their domain and ranges is the same. (Contributed by RP, 11-Jun-2020.) |
Ref | Expression |
---|---|
relexp0eq |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relexp0g 13162 |
. . 3
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2 | relexp0g 13162 |
. . 3
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3 | 1, 2 | eqeqan12d 2487 |
. 2
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4 | dfcleq 2465 |
. . . 4
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5 | alcom 1940 |
. . . . 5
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6 | 19.3v 1821 |
. . . . 5
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7 | ax6ev 1815 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() | |
8 | pm5.5 343 |
. . . . . . . . 9
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9 | 7, 8 | ax-mp 5 |
. . . . . . . 8
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10 | 19.23v 1826 |
. . . . . . . 8
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11 | 19.3v 1821 |
. . . . . . . 8
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12 | 9, 10, 11 | 3bitr4ri 286 |
. . . . . . 7
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13 | pm5.32 648 |
. . . . . . . . 9
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14 | ancom 457 |
. . . . . . . . . 10
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15 | ancom 457 |
. . . . . . . . . 10
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16 | 14, 15 | bibi12i 322 |
. . . . . . . . 9
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17 | 13, 16 | bitri 257 |
. . . . . . . 8
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18 | 17 | albii 1699 |
. . . . . . 7
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19 | 12, 18 | bitri 257 |
. . . . . 6
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20 | 19 | albii 1699 |
. . . . 5
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21 | 5, 6, 20 | 3bitr3i 283 |
. . . 4
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22 | 4, 21 | bitri 257 |
. . 3
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23 | eqopab2b 4731 |
. . 3
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24 | opabresid 5164 |
. . . 4
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25 | opabresid 5164 |
. . . 4
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26 | 24, 25 | eqeq12i 2485 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 22, 23, 26 | 3bitr2i 281 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 3, 27 | syl6rbbr 272 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 ax-1cn 9615 ax-icn 9616 ax-addcl 9617 ax-mulcl 9619 ax-i2m1 9625 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-sbc 3256 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-op 3966 df-uni 4191 df-br 4396 df-opab 4455 df-id 4754 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-iota 5553 df-fun 5591 df-fv 5597 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-n0 10894 df-relexp 13161 |
This theorem is referenced by: iunrelexp0 36365 |
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