Theorem prid2 3814

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 3813	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3783	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2313	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2209  Vcvv 2821  {cpr 3706

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712

This theorem is referenced by:  prel12  3891  opi2  4368  opeluu  4591  ontr2exmid  4667  onsucelsucexmid  4672  regexmidlemm  4674  ordtri2or2exmid  4713  ontri2orexmidim  4714  dmrnssfld  5040  funopg  5406  acexmidlema  6066  acexmidlemcase  6070  acexmidlem2  6072  1lt2o  6705  2dom  7083  en2m  7103  unfiexmid  7215  djuss  7400  pr2cv1  7531  exmidonfinlem  7535  exmidfodomrlemr  7544  exmidfodomrlemrALT  7545  exmidaclem  7554  cnelprrecn  8305  mnfxr  8372  sup3exmid  9277  m1expcl2  10976  fun2dmnop0  11280  fnpr2ob  13638  lgsdir2lem3  16063  upgrex  16258  upgr1een  16279  eulerpathprum  16635  bdop  16815  2o01f  16938  iswomni0  17006