Theorem vdif0 2328

Description: Universal class equality in terms of empty difference.

Assertion

Ref	Expression
vdif0	|- (A = V <-> (V \ A) = (/))

Proof of Theorem vdif0

Step	Hyp	Ref	Expression
1		vss 2307	. 2 |- (V (_ A <-> A = V)
2		ssdif0 2327	. 2 |- (V (_ A <-> (V \ A) = (/))
3	1, 2	bitr3 175	1 |- (A = V <-> (V \ A) = (/))

Syntax hints:   <-> wb 146   = wceq 956  Vcvv 1811   \ cdif 2044   (_ wss 2047  (/)c0 2280

This theorem is referenced by:  setind 4648

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281

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