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Mirrors > Home > ILE Home > Th. List > disjsn | Unicode version |
Description: Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.) |
Ref | Expression |
---|---|
disjsn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disj1 3473 |
. 2
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2 | con2b 669 |
. . . 4
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3 | velsn 3609 |
. . . . 5
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4 | 3 | imbi1i 238 |
. . . 4
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5 | imnan 690 |
. . . 4
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6 | 2, 4, 5 | 3bitri 206 |
. . 3
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7 | 6 | albii 1470 |
. 2
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8 | alnex 1499 |
. . 3
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9 | df-clel 2173 |
. . 3
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10 | 8, 9 | xchbinxr 683 |
. 2
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11 | 1, 7, 10 | 3bitri 206 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-dif 3131 df-in 3135 df-nul 3423 df-sn 3598 |
This theorem is referenced by: disjsn2 3655 ssdifsn 3720 orddisj 4545 ndmima 5005 funtpg 5267 fnunsn 5323 ressnop0 5697 ftpg 5700 fsnunf 5716 fsnunfv 5717 enpr2d 6816 phpm 6864 fiunsnnn 6880 ac6sfi 6897 unsnfi 6917 tpfidisj 6926 iunfidisj 6944 pm54.43 7188 dju1en 7211 fzpreddisj 10070 fzp1disj 10079 frecfzennn 10425 hashunsng 10786 hashxp 10805 fsumsplitsn 11417 sumtp 11421 fsumsplitsnun 11426 fsum2dlemstep 11441 fsumconst 11461 fsumabs 11472 fsumiun 11484 fprodm1 11605 fprodunsn 11611 fprod2dlemstep 11629 fprodsplitsn 11640 ennnfonelemhf1o 12413 structcnvcnv 12477 fsumcncntop 14026 |
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