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Theorem dfss7 4202
 Description: Alternate definition of subclass relationship. (Contributed by AV, 1-Aug-2022.)
Assertion
Ref Expression
dfss7 (𝐵𝐴 ↔ {𝑥𝐴𝑥𝐵} = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfss7
StepHypRef Expression
1 df-ss 3936 . 2 (𝐵𝐴 ↔ (𝐵𝐴) = 𝐵)
2 incom 4163 . . . 4 (𝐵𝐴) = (𝐴𝐵)
3 dfin5 3927 . . . 4 (𝐴𝐵) = {𝑥𝐴𝑥𝐵}
42, 3eqtri 2847 . . 3 (𝐵𝐴) = {𝑥𝐴𝑥𝐵}
54eqeq1i 2829 . 2 ((𝐵𝐴) = 𝐵 ↔ {𝑥𝐴𝑥𝐵} = 𝐵)
61, 5bitri 278 1 (𝐵𝐴 ↔ {𝑥𝐴𝑥𝐵} = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2115  {crab 3137   ∩ cin 3918   ⊆ wss 3919 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-rab 3142  df-in 3926  df-ss 3936 This theorem is referenced by:  qusker  30961  f1oresf1orab  43798
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