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Theorem brcnvssr 34579
 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 34573 . . 3 Rel S
21relbrcnv 5664 . 2 (𝐴 S 𝐵𝐵 S 𝐴)
3 brssr 34574 . 2 (𝐴𝑉 → (𝐵 S 𝐴𝐵𝐴))
42, 3syl5bb 272 1 (𝐴𝑉 → (𝐴 S 𝐵𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 2139   ⊆ wss 3715   class class class wbr 4804  ◡ccnv 5265   S cssr 34299 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-ssr 34571 This theorem is referenced by:  brcnvssrid  34580  br1cossxrncnvssrres  34581  dfcnvrefrels2  34599  dfcnvrefrels3  34600
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