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Theorem op2ndb 6180
Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5429 to extract the first member, op2nda 6181 for an alternate version, and op2nd 7931 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
Hypotheses
Ref Expression
cnvsn.1 𝐴 ∈ V
cnvsn.2 𝐵 ∈ V
Assertion
Ref Expression
op2ndb {⟨𝐴, 𝐵⟩} = 𝐵

Proof of Theorem op2ndb
StepHypRef Expression
1 cnvsn.1 . . . . . . 7 𝐴 ∈ V
2 cnvsn.2 . . . . . . 7 𝐵 ∈ V
31, 2cnvsn 6179 . . . . . 6 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
43inteqi 4912 . . . . 5 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
5 opex 5422 . . . . . 6 𝐵, 𝐴⟩ ∈ V
65intsn 4948 . . . . 5 {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴
74, 6eqtri 2765 . . . 4 {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴
87inteqi 4912 . . 3 {⟨𝐴, 𝐵⟩} = 𝐵, 𝐴
98inteqi 4912 . 2 {⟨𝐴, 𝐵⟩} = 𝐵, 𝐴
102, 1op1stb 5429 . 2 𝐵, 𝐴⟩ = 𝐵
119, 10eqtri 2765 1 {⟨𝐴, 𝐵⟩} = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  Vcvv 3446  {csn 4587  cop 4593   cint 4908  ccnv 5633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-int 4909  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642
This theorem is referenced by:  2ndval2  7940
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