Theorem vss 2311

Description: Only the universal class has the universal class as a subclass.

Assertion

Ref	Expression
vss	⊢ (V ⊆ A ↔ A = V)

Proof of Theorem vss

Step	Hyp	Ref	Expression
1		ssv 2084	. . . 4 ⊢ A ⊆ V
2	1	jctl 290	. . 3 ⊢ (V ⊆ A → (A ⊆ V ⋀ V ⊆ A))
3		eqss 2080	. . 3 ⊢ (A = V ↔ (A ⊆ V ⋀ V ⊆ A))
4	2, 3	sylibr 200	. 2 ⊢ (V ⊆ A → A = V)
5		eqimss2 2113	. 2 ⊢ (A = V → V ⊆ A)
6	4, 5	impbi 157	1 ⊢ (V ⊆ A ↔ A = V)

Syntax hints:   ↔ wb 146   ⋀ wa 223   = wceq 958  Vcvv 1814   ⊆ wss 2050

This theorem is referenced by:  vdif0 2332  dmen 4413

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-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054  df-ss 2056

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