Theorem inv1 3501

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 3397	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3217	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3219	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 3400	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3213	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1373  Vcvv 2773   ∩ cin 3169

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-ss 3183

This theorem is referenced by:  rint0  3930  riin0  4005  xpssres  5003  resdmdfsn  5011  imainrect  5137  xpima2m  5139  dmresv  5150