Theorem issetssr 38499

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 38498	. 2 ⊢ (𝐴 ∈ V → 𝐴 S 𝐴)
2		relssr 38496	. . 3 ⊢ Rel S
3	2	brrelex1i 5749	. 2 ⊢ (𝐴 S 𝐴 → 𝐴 ∈ V)
4	1, 3	impbii 209	1 ⊢ (𝐴 ∈ V ↔ 𝐴 S 𝐴)

Syntax hints:   ↔ wb 206   ∈ wcel 2108  Vcvv 3481   class class class wbr 5151   S cssr 38179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-br 5152  df-opab 5214  df-xp 5699  df-rel 5700  df-ssr 38494

This theorem is referenced by: (None)