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Theorem brcnvssr 38487
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 38481 . . 3 Rel S
21relbrcnv 6127 . 2 (𝐴 S 𝐵𝐵 S 𝐴)
3 brssr 38482 . 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 2105  wss 3962   class class class wbr 5147  ccnv 5687   S cssr 38164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-ssr 38479
This theorem is referenced by:  brcnvssrid  38488  br1cossxrncnvssrres  38489  dfcnvrefrels2  38509  dfcnvrefrels3  38510
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