Theorem inv1 3545

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 3441	. 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴
2		ssid 3258	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		ssv 3260	. . 3 ⊢ 𝐴 ⊆ V
4	2, 3	ssini 3444	. 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V)
5	1, 4	eqssi 3254	1 ⊢ (𝐴 ∩ V) = 𝐴

Syntax hints:   = wceq 1398  Vcvv 2813   ∩ cin 3210

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-ss 3224

This theorem is referenced by:  rint0  3988  riin0  4063  xpssres  5073  resdmdfsn  5081  imainrect  5208  xpima2m  5210  dmresv  5221