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Mirrors > Home > ILE Home > Th. List > prssg | 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 3703 | . . 3 | |
2 | snssg 3703 | . . 3 | |
3 | 1, 2 | bi2anan9 596 | . 2 |
4 | unss 3291 | . . 3 | |
5 | df-pr 3577 | . . . 4 | |
6 | 5 | sseq1i 3163 | . . 3 |
7 | 4, 6 | bitr4i 186 | . 2 |
8 | 3, 7 | bitrdi 195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2135 cun 3109 wss 3111 csn 3570 cpr 3571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 |
This theorem is referenced by: prssi 3725 prsspwg 3726 topgele 12574 |
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