Theorem inv1 3578

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 3476	. 2 ⊢ (A ∩ V) ⊆ A
2		ssid 3291	. . 3 ⊢ A ⊆ A
3		ssv 3292	. . 3 ⊢ A ⊆ V
4	2, 3	ssini 3479	. 2 ⊢ A ⊆ (A ∩ V)
5	1, 4	eqssi 3289	1 ⊢ (A ∩ V) = A

Syntax hints:   = wceq 1642  Vcvv 2860   ∩ cin 3209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  undif1  3626  dfif4  3674  rint0  3967  iinrab2  4030  riin0  4040  compldif  4070  xpkexg  4289