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Theorem intmin3 4895
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 3663 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
41, 3mpbiri 259 . 2 (𝐴𝑉𝐴 ∈ {𝑥𝜑})
5 intss1 4882 . 2 (𝐴 ∈ {𝑥𝜑} → {𝑥𝜑} ⊆ 𝐴)
64, 5syl 17 1 (𝐴𝑉 {𝑥𝜑} ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  {cab 2796  wss 3933   cint 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-in 3940  df-ss 3949  df-int 4868
This theorem is referenced by:  intabs  5236  intid  5341
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