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Theorem intmin3 4914
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 3612 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
41, 3mpbiri 259 . 2 (𝐴𝑉𝐴 ∈ {𝑥𝜑})
5 intss1 4901 . 2 (𝐴 ∈ {𝑥𝜑} → {𝑥𝜑} ⊆ 𝐴)
64, 5syl 17 1 (𝐴𝑉 {𝑥𝜑} ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wcel 2104  {cab 2713  wss 3892   cint 4886
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3439  df-in 3899  df-ss 3909  df-int 4887
This theorem is referenced by:  intabs  5275  intid  5382
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