Theorem inv 1723

Description: The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231.

Assertion

Ref	Expression
inv	⊢ (A ∩ V) = A

Proof of Theorem inv

Step	Hyp	Ref	Expression
1		inss1 1657	. 2 ⊢ (A ∩ V) ⊆ A
2		ssid 1519	. . 3 ⊢ A ⊆ A
3		ssv 1520	. . 3 ⊢ A ⊆ V
4	2, 3	ssini 1660	. 2 ⊢ A ⊆ (A ∩ V)
5	1, 4	eqssi 1517	1 ⊢ (A ∩ V) = A

Syntax hints:   = wceq 1091  Vcvv 1348   ∩ cin 1486

This theorem is referenced by:  undif1 1761

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492

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