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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssg 4327 | . . 3 | |
2 | snssg 4327 | . . 3 | |
3 | 1, 2 | bi2anan9 917 | . 2 |
4 | unss 3787 | . . 3 | |
5 | df-pr 4180 | . . . 4 | |
6 | 5 | sseq1i 3629 | . . 3 |
7 | 4, 6 | bitr4i 267 | . 2 |
8 | 3, 7 | syl6bb 276 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wcel 1990 cun 3572 wss 3574 csn 4177 cpr 4179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-un 3579 df-in 3581 df-ss 3588 df-sn 4178 df-pr 4180 |
This theorem is referenced by: prss 4351 prssi 4353 prsspwg 4355 ssprss 4356 prelpw 4914 hashdmpropge2 13265 lspprss 18992 lspvadd 19096 topgele 20734 umgredgprv 26002 usgredgprvALT 26087 dfnbgr2 26235 nbuhgr 26239 uhgrnbgr0nb 26250 2wlkdlem6 26827 1wlkdlem2 26998 dihmeetlem2N 36588 fourierdlem20 40344 fourierdlem50 40373 fourierdlem54 40377 fourierdlem64 40387 fourierdlem76 40399 omeunle 40730 |
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