Theorem inv1 3559

Description: The intersection of a class with the universal class is itself. Dual of un0 3556. 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 3451	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3268	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3270	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 3454	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3264	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1402  Vcvv 2821   ∩ cin 3219

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233

This theorem is referenced by:  vvin  3568  rint0  4004  riin0  4079  xpssres  5093  resdmdfsn  5101  imainrect  5228  xpima2m  5230  dmresv  5241