Theorem inv1 3404

Description: The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)

Assertion

Ref	Expression
inv1	⊢ (𝐴 ∩ V) = 𝐴

Proof of Theorem inv1

Step	Hyp	Ref	Expression
1		inss1 3301	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3122	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3124	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 3304	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3118	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1332  Vcvv 2689   ∩ cin 3075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089

This theorem is referenced by:  rint0  3818  riin0  3892  xpssres  4862  resdmdfsn  4870  imainrect  4992  xpima2m  4994  dmresv  5005