Theorem inv1 3247

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 3151	. 2 ⊢ (A ∩ V) ⊆ A
2		ssid 2958	. . 3 ⊢ A ⊆ A
3		ssv 2959	. . 3 ⊢ A ⊆ V
4	2, 3	ssini 3154	. 2 ⊢ A ⊆ (A ∩ V)
5	1, 4	eqssi 2955	1 ⊢ (A ∩ V) = A

Syntax hints:   = wceq 1242  Vcvv 2551   ∩ cin 2910

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925

This theorem is referenced by:  rint0  3645  riin0  3719  xpssres  4588  imainrect  4709  xpima2m  4711  dmresv  4722