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Theorem ssuniint 39564
 Description: Sufficient condition for being a subclass of the union of an intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypotheses
Ref Expression
ssuniint.x 𝑥𝜑
ssuniint.a (𝜑𝐴𝑉)
ssuniint.b ((𝜑𝑥𝐵) → 𝐴𝑥)
Assertion
Ref Expression
ssuniint (𝜑𝐴 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem ssuniint
StepHypRef Expression
1 ssuniint.x . . 3 𝑥𝜑
2 ssuniint.a . . 3 (𝜑𝐴𝑉)
3 ssuniint.b . . 3 ((𝜑𝑥𝐵) → 𝐴𝑥)
41, 2, 3elintd 39559 . 2 (𝜑𝐴 𝐵)
5 elssuni 4499 . 2 (𝐴 𝐵𝐴 𝐵)
64, 5syl 17 1 (𝜑𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  Ⅎwnf 1748   ∈ wcel 2030   ⊆ wss 3607  ∪ cuni 4468  ∩ cint 4507 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-in 3614  df-ss 3621  df-uni 4469  df-int 4508 This theorem is referenced by: (None)
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