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Theorem issetssr 34881
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 34880 . 2 (𝐴 ∈ V → 𝐴 S 𝐴)
2 relssr 34878 . . 3 Rel S
3 brrelex1 5403 . . 3 ((Rel S ∧ 𝐴 S 𝐴) → 𝐴 ∈ V)
42, 3mpan 680 . 2 (𝐴 S 𝐴𝐴 ∈ V)
51, 4impbii 201 1 (𝐴 ∈ V ↔ 𝐴 S 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wcel 2107  Vcvv 3398   class class class wbr 4886  Rel wrel 5360   S cssr 34609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-xp 5361  df-rel 5362  df-ssr 34876
This theorem is referenced by: (None)
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