Theorem inv1 3487

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 3383	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3203	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3205	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 3386	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3199	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1364  Vcvv 2763   ∩ cin 3156

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170

This theorem is referenced by:  rint0  3913  riin0  3988  xpssres  4981  resdmdfsn  4989  imainrect  5115  xpima2m  5117  dmresv  5128