Theorem inv1 3394

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 3291	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3112	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3114	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 3294	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3108	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1331  Vcvv 2681   ∩ cin 3065

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079

This theorem is referenced by:  rint0  3805  riin0  3879  xpssres  4849  resdmdfsn  4857  imainrect  4979  xpima2m  4981  dmresv  4992