Theorem prid2 3773

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 3772	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3742	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2304	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2200  Vcvv 2799  {cpr 3667

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673

This theorem is referenced by:  prel12  3849  opi2  4319  opeluu  4541  ontr2exmid  4617  onsucelsucexmid  4622  regexmidlemm  4624  ordtri2or2exmid  4663  ontri2orexmidim  4664  dmrnssfld  4987  funopg  5352  acexmidlema  5998  acexmidlemcase  6002  acexmidlem2  6004  1lt2o  6596  2dom  6966  en2m  6982  unfiexmid  7091  djuss  7248  pr2cv1  7379  exmidonfinlem  7382  exmidfodomrlemr  7391  exmidfodomrlemrALT  7392  exmidaclem  7401  cnelprrecn  8146  mnfxr  8214  sup3exmid  9115  m1expcl2  10795  fun2dmnop0  11082  fnpr2ob  13388  lgsdir2lem3  15724  upgrex  15918  bdop  16293  2o01f  16417  iswomni0  16479