Theorem issetssr 39087

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

Step	Hyp	Ref	Expression
1		brssrid 39086	. 2 ⊢ (𝐴 ∈ V → 𝐴 S 𝐴)
2		relssr 39084	. . 3 ⊢ Rel S
3	2	brrelex1i 5705	. 2 ⊢ (𝐴 S 𝐴 → 𝐴 ∈ V)
4	1, 3	impbii 211	1 ⊢ (𝐴 ∈ V ↔ 𝐴 S 𝐴)

Syntax hints:   ↔ wb 208   ∈ wcel 2144  Vcvv 3456   class class class wbr 5102   S cssr 38690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-ssr 39082

This theorem is referenced by: (None)