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Mirrors > Home > ILE Home > Th. List > eldju2ndl | Unicode version |
Description: The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022.) |
Ref | Expression |
---|---|
eldju2ndl |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 6931 |
. . . . 5
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2 | 1 | eleq2i 2207 |
. . . 4
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3 | elun 3222 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 2, 3 | bitri 183 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | elxp6 6075 |
. . . . 5
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6 | simprr 522 |
. . . . . 6
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7 | 6 | a1d 22 |
. . . . 5
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8 | 5, 7 | sylbi 120 |
. . . 4
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9 | elxp6 6075 |
. . . . 5
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10 | elsni 3550 |
. . . . . . 7
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11 | 1n0 6337 |
. . . . . . . 8
![]() ![]() ![]() ![]() | |
12 | neeq1 2322 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 11, 12 | mpbiri 167 |
. . . . . . 7
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14 | eqneqall 2319 |
. . . . . . . 8
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15 | 14 | com12 30 |
. . . . . . 7
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16 | 10, 13, 15 | 3syl 17 |
. . . . . 6
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17 | 16 | ad2antrl 482 |
. . . . 5
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18 | 9, 17 | sylbi 120 |
. . . 4
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19 | 8, 18 | jaoi 706 |
. . 3
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20 | 4, 19 | sylbi 120 |
. 2
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21 | 20 | imp 123 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-suc 4301 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fv 5139 df-1st 6046 df-2nd 6047 df-1o 6321 df-dju 6931 |
This theorem is referenced by: updjudhf 6972 subctctexmid 13369 |
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