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Mirrors > Home > MPE Home > Th. List > prssg | Structured version Visualization version Unicode version |
Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
prssg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssg 4118 |
. . 3
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2 | snssg 4118 |
. . 3
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3 | 1, 2 | bi2anan9 889 |
. 2
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4 | unss 3620 |
. . 3
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5 | df-pr 3983 |
. . . 4
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6 | 5 | sseq1i 3468 |
. . 3
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7 | 4, 6 | bitr4i 260 |
. 2
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8 | 3, 7 | syl6bb 269 |
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-or 376 df-an 377 df-3an 993 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-un 3421 df-in 3423 df-ss 3430 df-sn 3981 df-pr 3983 |
This theorem is referenced by: prssi 4141 prsspwg 4142 lspprss 18264 lspvadd 18368 topgele 19998 usgraedgprv 25152 usgraedgrnv 25153 usgraedg4 25163 2trllemH 25331 2trllemE 25332 dihmeetlem2N 34912 prssd 37422 fourierdlem20 38027 fourierdlem50 38058 fourierdlem54 38062 fourierdlem64 38072 fourierdlem76 38084 omeunle 38375 prelpw 39033 ssprss 39041 umgredgprv 39243 usgredgprvALT 39326 dfnbgr2 39457 nbuhgr 39461 uhgrnbgr0nb 39472 11wlkdlem2 39853 21wlkdlem6 39880 |
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