Theorem inv1 3338

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 3235	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3059	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3061	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 3238	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3055	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1296  Vcvv 2633   ∩ cin 3012

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026

This theorem is referenced by:  rint0  3749  riin0  3823  xpssres  4780  resdmdfsn  4788  imainrect  4910  xpima2m  4912  dmresv  4923