Theorem issetssr 38467

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 38466	. 2 ⊢ (𝐴 ∈ V → 𝐴 S 𝐴)
2		relssr 38464	. . 3 ⊢ Rel S
3	2	brrelex1i 5687	. 2 ⊢ (𝐴 S 𝐴 → 𝐴 ∈ V)
4	1, 3	impbii 209	1 ⊢ (𝐴 ∈ V ↔ 𝐴 S 𝐴)

Syntax hints:   ↔ wb 206   ∈ wcel 2109  Vcvv 3444   class class class wbr 5102   S cssr 38145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-ssr 38462

This theorem is referenced by: (None)