Theorem inv1 3528

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 3424	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3244	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3246	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 3427	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3240	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1395  Vcvv 2799   ∩ cin 3196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210

This theorem is referenced by:  rint0  3961  riin0  4036  xpssres  5039  resdmdfsn  5047  imainrect  5173  xpima2m  5175  dmresv  5186