Theorem difid 1755

Description: The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28.

Assertion

Ref	Expression
difid	⊢ (A ∖ A) = ∅

Proof of Theorem difid

Step	Hyp	Ref	Expression
1		ssid 1519	. 2 ⊢ A ⊆ A
2		ssdif0 1748	. 2 COLOR="#808080">⊢ (A ⊆ A ↔ (A ∖ A) = ∅)
31, 2	mpbi 164	1 ⊢ (A ∖ A) = ∅

Syntax hints:   = wceq 1091   ∖ cdif 1484   ⊆ wss 1487  ∅c0 1707

This theorem is referenced by:  dif0 1756  difin0 1759  difun2 1763

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-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708

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