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Theorem dfss5 3995
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 3721 . 2 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
2 clel5 3471 . . 3 (𝑥𝐵 ↔ ∃𝑦𝐵 𝑥 = 𝑦)
32ralbii 3106 . 2 (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 = 𝑦)
41, 3bitri 264 1 (𝐴𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2127  wral 3038  wrex 3039  wss 3703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-v 3330  df-in 3710  df-ss 3717
This theorem is referenced by:  usgrsscusgr  26537
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