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Theorem intmin3 3616
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 {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 2665 . . 3 (A 𝑉 → (A {xφ} ↔ ψ))
41, 3mpbiri 157 . 2 (A 𝑉A {xφ})
5 intss1 3604 . 2 (A {xφ} → {xφ} ⊆ A)
64, 5syl 14 1 (A 𝑉 {xφ} ⊆ A)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228   wcel 1374  {cab 2008  wss 2894   cint 3589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-int 3590
This theorem is referenced by:  intid  3934
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