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Theorem dfss7 40102
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 3557 . 2 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
2 clel5 40101 . . 3 (𝑥𝐵 ↔ ∃𝑦𝐵 𝑥 = 𝑦)
32ralbii 2962 . 2 (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 = 𝑦)
41, 3bitri 262 1 (𝐴𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wcel 1976  wral 2895  wrex 2896  wss 3539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-v 3174  df-in 3546  df-ss 3553
This theorem is referenced by:  usgrsscusgr  40671
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