Theorem inv1 3471

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 3367	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3187	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3189	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 3370	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3183	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1363  Vcvv 2749   ∩ cin 3140

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-in 3147  df-ss 3154

This theorem is referenced by:  rint0  3895  riin0  3970  xpssres  4954  resdmdfsn  4962  imainrect  5086  xpima2m  5088  dmresv  5099