Theorem vdif0 2332

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 2311	. 2 |- (V (_ A <-> A = V)
2		ssdif0 2331	. 2 |- (V (_ A <-> (V \ A) = (/))
3	1, 2	bitr3 175	1 |- (A = V <-> (V \ A) = (/))

Syntax hints:   <-> wb 146   = wceq 958  Vcvv 1814   \ cdif 2047   (_ wss 2050  (/)c0 2283

This theorem is referenced by:  setind 4658

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-nul 2284

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