Theorem prid2 3740

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 ∈ V

Assertion

Ref	Expression
prid2	⊢ 𝐵 ∈ {𝐴, 𝐵}

Proof of Theorem prid2

Step	Hyp	Ref	Expression
1		prid2.1	. . 3 ⊢ 𝐵 ∈ V
2	1	prid1 3739	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3709	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2280	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2176  Vcvv 2772  {cpr 3634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640

This theorem is referenced by:  prel12  3812  opi2  4277  opeluu  4497  ontr2exmid  4573  onsucelsucexmid  4578  regexmidlemm  4580  ordtri2or2exmid  4619  ontri2orexmidim  4620  dmrnssfld  4941  funopg  5305  acexmidlema  5935  acexmidlemcase  5939  acexmidlem2  5941  1lt2o  6528  2dom  6897  unfiexmid  7015  djuss  7172  exmidonfinlem  7301  exmidfodomrlemr  7310  exmidfodomrlemrALT  7311  exmidaclem  7320  cnelprrecn  8061  mnfxr  8129  sup3exmid  9030  m1expcl2  10706  fun2dmnop0  10992  fnpr2ob  13172  lgsdir2lem3  15507  bdop  15811  2o01f  15931  iswomni0  15990