Theorem prssg 4185

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.)

Assertion

Ref	Expression
prssg

Proof of Theorem prssg

Step	Hyp	Ref	Expression
1		snssg 4163	. . 3
2		snssg 4163	. . 3
3	1, 2	bi2anan9 873	. 2
4		unss 3677	. . 3
5		df-pr 4032	. . . 4
6	5	sseq1i 3527	. . 3
7	4, 6	bitr4i 252	. 2
8	3, 7	syl6bb 261	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  e.wcel 1818  u.cun 3473  C_wss 3475  {csn 4029  {cpr 4031

This theorem is referenced by:  prssi  4186  prsspwg  4187  lspprss  17638  lspvadd  17742  topgele  19435  usgraedgprv  24376  usgraedgrnv  24377  usgraedg4  24387  2trllemH  24554  2trllemE  24555  fourierdlem20  31909  fourierdlem50  31939  fourierdlem54  31943  fourierdlem64  31953  fourierdlem76  31965  prelpw  32299  dihmeetlem2N  37026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-in 3482  df-ss 3489  df-sn 4030  df-pr 4032