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Mirrors > Home > MPE Home > Th. List > mpt2xopxnop0 | Structured version Visualization version Unicode version |
Description: If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is not an ordered pair, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.) |
Ref | Expression |
---|---|
mpt2xopn0yelv.f |
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Ref | Expression |
---|---|
mpt2xopxnop0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neq0 3733 |
. . 3
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2 | mpt2xopn0yelv.f |
. . . . . . 7
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3 | 2 | dmmpt2ssx 6877 |
. . . . . 6
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4 | elfvdm 5905 |
. . . . . . 7
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5 | df-ov 6311 |
. . . . . . 7
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6 | 4, 5 | eleq2s 2567 |
. . . . . 6
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7 | 3, 6 | sseldi 3416 |
. . . . 5
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8 | fveq2 5879 |
. . . . . . 7
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9 | 8 | opeliunxp2 4978 |
. . . . . 6
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10 | eluni 4193 |
. . . . . . . . 9
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11 | ne0i 3728 |
. . . . . . . . . . . . 13
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12 | 11 | ad2antlr 741 |
. . . . . . . . . . . 12
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13 | dmsnn0 5308 |
. . . . . . . . . . . 12
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14 | 12, 13 | sylibr 217 |
. . . . . . . . . . 11
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15 | 14 | ex 441 |
. . . . . . . . . 10
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16 | 15 | exlimiv 1784 |
. . . . . . . . 9
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17 | 10, 16 | sylbi 200 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 1stval 6814 |
. . . . . . . 8
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19 | 17, 18 | eleq2s 2567 |
. . . . . . 7
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20 | 19 | impcom 437 |
. . . . . 6
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21 | 9, 20 | sylbi 200 |
. . . . 5
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22 | 7, 21 | syl 17 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | exlimiv 1784 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 1, 23 | sylbi 200 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | con1i 134 |
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 |
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-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-op 3966 df-uni 4191 df-iun 4271 df-br 4396 df-opab 4455 df-mpt 4456 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-ima 4852 df-iota 5553 df-fun 5591 df-fv 5597 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-1st 6812 df-2nd 6813 |
This theorem is referenced by: mpt2xopx0ov0 6981 mpt2xopxprcov0 6982 |
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