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Theorem issetssr 38408
Description: Two ways of expressing set existence. (Contributed by Peter Mazsa, 1-Aug-2019.)
Assertion
Ref Expression
issetssr (𝐴 ∈ V ↔ 𝐴 S 𝐴)

Proof of Theorem issetssr
StepHypRef Expression
1 brssrid 38407 . 2 (𝐴 ∈ V → 𝐴 S 𝐴)
2 relssr 38405 . . 3 Rel S
32brrelex1i 5755 . 2 (𝐴 S 𝐴𝐴 ∈ V)
41, 3impbii 209 1 (𝐴 ∈ V ↔ 𝐴 S 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2103  Vcvv 3482   class class class wbr 5169   S cssr 38087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-xp 5705  df-rel 5706  df-ssr 38403
This theorem is referenced by: (None)
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