Theorem issetssr 39260

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 39259	. 2 ⊢ (𝐴 ∈ V → 𝐴 S 𝐴)
2		relssr 39257	. . 3 ⊢ Rel S
3	2	brrelex1i 5717	. 2 ⊢ (𝐴 S 𝐴 → 𝐴 ∈ V)
4	1, 3	impbii 212	1 ⊢ (𝐴 ∈ V ↔ 𝐴 S 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2143  Vcvv 3455   class class class wbr 5109   S cssr 38863

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-ssr 39255

This theorem is used by: (None)