Theorem prid2 3468

Description: An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
prid2.1	|- B e. _V

Assertion

Ref	Expression
prid2	|- B e. {A , B }

Proof of Theorem prid2

Step	Hyp	Ref	Expression
1		prid2.1	. . 3 |- B e. _V
2	1	prid1 3467	. 2 |- B e. {B , A }
3		prcom 3437	. 2 |- {B , A } = {A , B }
4	2, 3	eleqtri 2109	1 |- B e. {A , B }

Syntax hints:   e. wcel 1390   _Vcvv 2551   {cpr 3368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374

This theorem is referenced by:  prel12  3533  opi2  3961  opeluu  4148  onsucelsucexmid  4215  regexmidlemm  4217  dmrnssfld  4538  funopg  4877  acexmidlema  5446  acexmidlemcase  5450  acexmidlem2  5452  2dom  6221  cnelprrecn  6815  mnfxr  8464  m1expcl2  8931  bdop  9330