Theorem inv1 3496

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 3392	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3212	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3214	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 3395	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3208	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1372  Vcvv 2771   ∩ cin 3164

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178

This theorem is referenced by:  rint0  3923  riin0  3998  xpssres  4991  resdmdfsn  4999  imainrect  5125  xpima2m  5127  dmresv  5138