Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brcnvssr Structured version   Visualization version   GIF version

Theorem brcnvssr 38507
Description: The converse of a subset relation swaps arguments. (Contributed by Peter Mazsa, 1-Aug-2019.)
Assertion
Ref Expression
brcnvssr (𝐴𝑉 → (𝐴 S 𝐵𝐵𝐴))

Proof of Theorem brcnvssr
StepHypRef Expression
1 relssr 38501 . . 3 Rel S
21relbrcnv 6125 . 2 (𝐴 S 𝐵𝐵 S 𝐴)
3 brssr 38502 . 2 (𝐴𝑉 → (𝐵 S 𝐴𝐵𝐴))
42, 3bitrid 283 1 (𝐴𝑉 → (𝐴 S 𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2108  wss 3951   class class class wbr 5143  ccnv 5684   S cssr 38185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-ssr 38499
This theorem is referenced by:  brcnvssrid  38508  br1cossxrncnvssrres  38509  dfcnvrefrels2  38529  dfcnvrefrels3  38530
  Copyright terms: Public domain W3C validator