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Theorem dfss7 40304
 Description: Alternate definition of subclass relationship: a class 𝐴 is a subclass of another class 𝐵 iff each element of 𝐴 is equal to an element of 𝐵. (Contributed by AV, 13-Nov-2020.)
Assertion
Ref Expression
dfss7 (𝐴𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 = 𝑦)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵,𝑦
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem dfss7
StepHypRef Expression
1 dfss3 3558 . 2 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
2 clel5 40303 . . 3 (𝑥𝐵 ↔ ∃𝑦𝐵 𝑥 = 𝑦)
32ralbii 2963 . 2 (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 = 𝑦)
41, 3bitri 263 1 (𝐴𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554 This theorem is referenced by:  usgrsscusgr  40676
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