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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 3352 |
. 2
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2 | con2b 631 |
. . . 4
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3 | velsn 3483 |
. . . . 5
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4 | 3 | imbi1i 237 |
. . . 4
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5 | imnan 662 |
. . . 4
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6 | 2, 4, 5 | 3bitri 205 |
. . 3
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7 | 6 | albii 1411 |
. 2
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8 | alnex 1440 |
. . 3
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9 | df-clel 2091 |
. . 3
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10 | 8, 9 | xchbinxr 646 |
. 2
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11 | 1, 7, 10 | 3bitri 205 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-v 2635 df-dif 3015 df-in 3019 df-nul 3303 df-sn 3472 |
This theorem is referenced by: disjsn2 3525 ssdifsn 3590 orddisj 4390 ndmima 4842 funtpg 5099 fnunsn 5155 ressnop0 5517 ftpg 5520 fsnunf 5536 fsnunfv 5537 phpm 6661 fiunsnnn 6677 ac6sfi 6694 unsnfi 6709 tpfidisj 6718 iunfidisj 6735 pm54.43 6915 fzpreddisj 9634 fzp1disj 9643 frecfzennn 9982 hashunsng 10346 hashxp 10365 fsumsplitsn 10969 sumtp 10973 fsumsplitsnun 10978 fsum2dlemstep 10993 fsumconst 11013 fsumabs 11024 fsumiun 11036 structcnvcnv 11675 |
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