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Theorem dfss7 4065
 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 3806 . 2 (𝐵𝐴 ↔ (𝐵𝐴) = 𝐵)
2 incom 4028 . . . 4 (𝐵𝐴) = (𝐴𝐵)
3 dfin5 3800 . . . 4 (𝐴𝐵) = {𝑥𝐴𝑥𝐵}
42, 3eqtri 2802 . . 3 (𝐵𝐴) = {𝑥𝐴𝑥𝐵}
54eqeq1i 2783 . 2 ((𝐵𝐴) = 𝐵 ↔ {𝑥𝐴𝑥𝐵} = 𝐵)
61, 5bitri 267 1 (𝐵𝐴 ↔ {𝑥𝐴𝑥𝐵} = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1601   ∈ wcel 2107  {crab 3094   ∩ cin 3791   ⊆ wss 3792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-in 3799  df-ss 3806 This theorem is referenced by:  qusker  30411  f1oresf1orab  42340
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