ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opswapg Structured version   GIF version

Theorem opswapg 4749
Description: Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.)
Assertion
Ref Expression
opswapg ((A 𝑉 B 𝑊) → {⟨A, B⟩} = ⟨B, A⟩)

Proof of Theorem opswapg
StepHypRef Expression
1 cnvsng 4748 . . 3 ((A 𝑉 B 𝑊) → {⟨A, B⟩} = {⟨B, A⟩})
21unieqd 3581 . 2 ((A 𝑉 B 𝑊) → {⟨A, B⟩} = {⟨B, A⟩})
3 elex 2560 . . . 4 (B 𝑊B V)
4 elex 2560 . . . 4 (A 𝑉A V)
5 opexgOLD 3955 . . . 4 ((B V A V) → ⟨B, A V)
63, 4, 5syl2anr 274 . . 3 ((A 𝑉 B 𝑊) → ⟨B, A V)
7 unisng 3587 . . 3 (⟨B, A V → {⟨B, A⟩} = ⟨B, A⟩)
86, 7syl 14 . 2 ((A 𝑉 B 𝑊) → {⟨B, A⟩} = ⟨B, A⟩)
92, 8eqtrd 2069 1 ((A 𝑉 B 𝑊) → {⟨A, B⟩} = ⟨B, A⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  {csn 3366  cop 3369   cuni 3570  ccnv 4286
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-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934
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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-br 3755  df-opab 3809  df-xp 4293  df-rel 4294  df-cnv 4295
This theorem is referenced by:  2nd1st  5745  cnvf1olem  5784  brtposg  5807  dftpos4  5816  tpostpos  5817  xpcomco  6229
  Copyright terms: Public domain W3C validator