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Theorem op2ndb 4747
Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4175 to extract the first member and op2nda 4748 for an alternate 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 4746 . . . . . 6 {⟨A, B⟩} = {⟨B, A⟩}
43inteqi 3610 . . . . 5 {⟨A, B⟩} = {⟨B, A⟩}
5 opexgOLD 3956 . . . . . . 7 ((B V A V) → ⟨B, A V)
62, 1, 5mp2an 402 . . . . . 6 B, A V
76intsn 3641 . . . . 5 {⟨B, A⟩} = ⟨B, A
84, 7eqtri 2057 . . . 4 {⟨A, B⟩} = ⟨B, A
98inteqi 3610 . . 3 {⟨A, B⟩} = B, A
109inteqi 3610 . 2 {⟨A, B⟩} = B, A
112, 1op1stb 4175 . 2 B, A⟩ = B
1210, 11eqtri 2057 1 {⟨A, B⟩} = B
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  Vcvv 2551  {csn 3367  cop 3370   cint 3606  ccnv 4287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-int 3607  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296
This theorem is referenced by:  2ndval2  5725
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