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Theorem intmin3 4440
 Description: Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members. (Contributed by NM, 3-Jul-2005.)
Hypotheses
Ref Expression
intmin3.2 (𝑥 = 𝐴 → (𝜑𝜓))
intmin3.3 𝜓
Assertion
Ref Expression
intmin3 (𝐴𝑉 {𝑥𝜑} ⊆ 𝐴)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem intmin3
StepHypRef Expression
1 intmin3.3 . . 3 𝜓
2 intmin3.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
32elabg 3320 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
41, 3mpbiri 247 . 2 (𝐴𝑉𝐴 ∈ {𝑥𝜑})
5 intss1 4427 . 2 (𝐴 ∈ {𝑥𝜑} → {𝑥𝜑} ⊆ 𝐴)
64, 5syl 17 1 (𝐴𝑉 {𝑥𝜑} ⊆ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596   ⊆ wss 3540  ∩ cint 4410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-int 4411 This theorem is referenced by:  intabs  4752  intid  4853
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