Theorem inv1 3445

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 3342	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3162	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3164	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 3345	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3158	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1343  Vcvv 2726   ∩ cin 3115

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129

This theorem is referenced by:  rint0  3863  riin0  3937  xpssres  4919  resdmdfsn  4927  imainrect  5049  xpima2m  5051  dmresv  5062