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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 3475 |
. 2
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2 | con2b 669 |
. . . 4
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3 | velsn 3611 |
. . . . 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 2741 df-dif 3133 df-in 3137 df-nul 3425 df-sn 3600 |
This theorem is referenced by: disjsn2 3657 ssdifsn 3722 orddisj 4547 ndmima 5007 funtpg 5269 fnunsn 5325 ressnop0 5699 ftpg 5702 fsnunf 5718 fsnunfv 5719 enpr2d 6819 phpm 6867 fiunsnnn 6883 ac6sfi 6900 unsnfi 6920 tpfidisj 6929 iunfidisj 6947 pm54.43 7191 dju1en 7214 fzpreddisj 10073 fzp1disj 10082 frecfzennn 10428 hashunsng 10789 hashxp 10808 fsumsplitsn 11420 sumtp 11424 fsumsplitsnun 11429 fsum2dlemstep 11444 fsumconst 11464 fsumabs 11475 fsumiun 11487 fprodm1 11608 fprodunsn 11614 fprod2dlemstep 11632 fprodsplitsn 11643 ennnfonelemhf1o 12416 structcnvcnv 12480 fsumcncntop 14095 |
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