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Mirrors > Home > MPE Home > Th. List > sneqr | Structured version Visualization version Unicode version |
Description: If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.) |
Ref | Expression |
---|---|
sneqr.1 |
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Ref | Expression |
---|---|
sneqr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneqr.1 |
. . . 4
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2 | 1 | snid 4008 |
. . 3
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3 | eleq2 2529 |
. . 3
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4 | 2, 3 | mpbii 216 |
. 2
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5 | 1 | elsnc 4004 |
. 2
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6 | 4, 5 | sylib 201 |
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 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 |
This theorem depends on definitions: df-bi 190 df-an 377 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-v 3059 df-sn 3981 |
This theorem is referenced by: snsssn 4153 sneqrg 4154 opth1 4692 opthwiener 4720 canth2 7756 axcc2lem 8897 hashge3el3dif 12681 dis2ndc 20530 axlowdim1 25045 bj-snsetex 31603 poimirlem13 31999 poimirlem14 32000 wopprc 35931 hoidmv1le 38523 propeqop 39136 funsndifnop 39160 |
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