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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 13097 |
. . 3
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2 | relexp0g 13097 |
. . 3
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3 | 1, 2 | eqeqan12d 2469 |
. 2
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4 | dfcleq 2447 |
. . . 4
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5 | alcom 1925 |
. . . . 5
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6 | 19.3v 1815 |
. . . . 5
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7 | ax6ev 1809 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() | |
8 | pm5.5 338 |
. . . . . . . . 9
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9 | 7, 8 | ax-mp 5 |
. . . . . . . 8
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10 | 19.23v 1820 |
. . . . . . . 8
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11 | 19.3v 1815 |
. . . . . . . 8
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12 | 9, 10, 11 | 3bitr4ri 282 |
. . . . . . 7
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13 | pm5.32 642 |
. . . . . . . . 9
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14 | ancom 452 |
. . . . . . . . . 10
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15 | ancom 452 |
. . . . . . . . . 10
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16 | 14, 15 | bibi12i 317 |
. . . . . . . . 9
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17 | 13, 16 | bitri 253 |
. . . . . . . 8
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18 | 17 | albii 1693 |
. . . . . . 7
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19 | 12, 18 | bitri 253 |
. . . . . 6
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20 | 19 | albii 1693 |
. . . . 5
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21 | 5, 6, 20 | 3bitr3i 279 |
. . . 4
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22 | 4, 21 | bitri 253 |
. . 3
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23 | eqopab2b 4734 |
. . 3
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24 | opabresid 5161 |
. . . 4
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25 | opabresid 5161 |
. . . 4
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26 | 24, 25 | eqeq12i 2467 |
. . 3
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27 | 22, 23, 26 | 3bitr2i 277 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 3, 27 | syl6rbbr 268 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 ax-1cn 9602 ax-icn 9603 ax-addcl 9604 ax-mulcl 9606 ax-i2m1 9612 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-rab 2748 df-v 3049 df-sbc 3270 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-br 4406 df-opab 4465 df-id 4752 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-iota 5549 df-fun 5587 df-fv 5593 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-n0 10877 df-relexp 13096 |
This theorem is referenced by: iunrelexp0 36306 |
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