Theorem inv1 3253

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 3157	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 2964	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 2965	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 3160	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 2961	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1243  Vcvv 2557   ∩ cin 2916

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931

This theorem is referenced by:  rint0  3654  riin0  3728  xpssres  4645  imainrect  4766  xpima2m  4768  dmresv  4779