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Theorem dfss5 4171
 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
dfss5 (𝐴𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 = 𝑦)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵,𝑦
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem dfss5
StepHypRef Expression
1 dfss3 3882 . 2 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
2 clel5 3580 . . 3 (𝑥𝐵 ↔ ∃𝑦𝐵 𝑥 = 𝑦)
32ralbii 3097 . 2 (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 = 𝑦)
41, 3bitri 278 1 (𝐴𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071   ⊆ wss 3860 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-in 3867  df-ss 3877 This theorem is referenced by:  usgrsscusgr  27354
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