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Theorem ssunib 42424
Description: Two ways to say a class is a subclass of a union. (Contributed by RP, 27-Jan-2025.)
Assertion
Ref Expression
ssunib (𝐴 𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵,𝑦
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem ssunib
StepHypRef Expression
1 dfss3 3962 . 2 (𝐴 𝐵 ↔ ∀𝑥𝐴 𝑥 𝐵)
2 eluni2 4903 . . 3 (𝑥 𝐵 ↔ ∃𝑦𝐵 𝑥𝑦)
32ralbii 3085 . 2 (∀𝑥𝐴 𝑥 𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
41, 3bitri 275 1 (𝐴 𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2098  wral 3053  wrex 3062  wss 3940   cuni 4899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-v 3468  df-in 3947  df-ss 3957  df-uni 4900
This theorem is referenced by:  onmaxnelsup  42427  onsupnmax  42432
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