Theorem prid2 3721

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 3720	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3690	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2264	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2160  Vcvv 2756  {cpr 3615

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2758  df-un 3153  df-sn 3620  df-pr 3621

This theorem is referenced by:  prel12  3793  opi2  4258  opeluu  4475  ontr2exmid  4549  onsucelsucexmid  4554  regexmidlemm  4556  ordtri2or2exmid  4595  ontri2orexmidim  4596  dmrnssfld  4915  funopg  5276  acexmidlema  5897  acexmidlemcase  5901  acexmidlem2  5903  1lt2o  6482  2dom  6846  unfiexmid  6961  djuss  7115  exmidonfinlem  7239  exmidfodomrlemr  7248  exmidfodomrlemrALT  7249  exmidaclem  7254  cnelprrecn  7994  mnfxr  8062  sup3exmid  8962  m1expcl2  10606  fnpr2ob  12897  lgsdir2lem3  15074  bdop  15291  2o01f  15411  iswomni0  15465