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Theorem intmin3 3954
 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 (A V{x φ} 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 2986 . . 3 (A V → (A {x φ} ↔ ψ))
41, 3mpbiri 224 . 2 (A VA {x φ})
5 intss1 3941 . 2 (A {x φ} → {x φ} A)
64, 5syl 15 1 (A V{x φ} A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {cab 2339   ⊆ wss 3257  ∩cint 3926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-int 3927 This theorem is referenced by: (None)
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