Theorem inv1 3531

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 3427	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3247	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3249	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 3430	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3243	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1397  Vcvv 2802   ∩ cin 3199

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213

This theorem is referenced by:  rint0  3967  riin0  4042  xpssres  5048  resdmdfsn  5056  imainrect  5182  xpima2m  5184  dmresv  5195