Theorem issetssr 38959

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 38958	. 2 ⊢ (𝐴 ∈ V → 𝐴 S 𝐴)
2		relssr 38956	. . 3 ⊢ Rel S
3	2	brrelex1i 5675	. 2 ⊢ (𝐴 S 𝐴 → 𝐴 ∈ V)
4	1, 3	impbii 210	1 ⊢ (𝐴 ∈ V ↔ 𝐴 S 𝐴)

Syntax hints:   ↔ wb 207   ∈ wcel 2119  Vcvv 3431   class class class wbr 5073   S cssr 38562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5219  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-br 5074  df-opab 5136  df-xp 5625  df-rel 5626  df-ssr 38954

This theorem is referenced by: (None)