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Theorem ssuniint 45056
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 45052 . 2 (𝜑𝐴 𝐵)
5 elssuni 4935 . 2 (𝐴 𝐵𝐴 𝐵)
64, 5syl 17 1 (𝜑𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1783  wcel 2108  wss 3950   cuni 4905   cint 4944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-v 3481  df-ss 3967  df-uni 4906  df-int 4945
This theorem is referenced by: (None)
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