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Theorem brcnvssr 34797
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 34791 . . 3 Rel S
21relbrcnv 5747 . 2 (𝐴 S 𝐵𝐵 S 𝐴)
3 brssr 34792 . 2 (𝐴𝑉 → (𝐵 S 𝐴𝐵𝐴))
42, 3syl5bb 275 1 (𝐴𝑉 → (𝐴 S 𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wcel 2164  wss 3798   class class class wbr 4873  ccnv 5341   S cssr 34520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-xp 5348  df-rel 5349  df-cnv 5350  df-ssr 34789
This theorem is referenced by:  brcnvssrid  34798  br1cossxrncnvssrres  34799  dfcnvrefrels2  34817  dfcnvrefrels3  34818
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