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Theorem issetssr 35783
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 35782 . 2 (𝐴 ∈ V → 𝐴 S 𝐴)
2 relssr 35780 . . 3 Rel S
32brrelex1i 5581 . 2 (𝐴 S 𝐴𝐴 ∈ V)
41, 3impbii 212 1 (𝐴 ∈ V ↔ 𝐴 S 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2115  Vcvv 3471   class class class wbr 5039   S cssr 35496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-xp 5534  df-rel 5535  df-ssr 35778
This theorem is referenced by: (None)
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