Theorem inv1 3451

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 3347	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3167	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3169	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 3350	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3163	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1348  Vcvv 2730   ∩ cin 3120

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134

This theorem is referenced by:  rint0  3870  riin0  3944  xpssres  4926  resdmdfsn  4934  imainrect  5056  xpima2m  5058  dmresv  5069