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Theorem ssdifeq0 3632
 Description: A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)
Assertion
Ref Expression
ssdifeq0 (A (B A) ↔ A = )

Proof of Theorem ssdifeq0
StepHypRef Expression
1 inidm 3464 . . 3 (AA) = A
2 ssdifin0 3631 . . 3 (A (B A) → (AA) = )
31, 2syl5eqr 2399 . 2 (A (B A) → A = )
4 0ss 3579 . . 3 (B )
5 id 19 . . . 4 (A = A = )
6 difeq2 3247 . . . 4 (A = → (B A) = (B ))
75, 6sseq12d 3300 . . 3 (A = → (A (B A) ↔ (B )))
84, 7mpbiri 224 . 2 (A = A (B A))
93, 8impbii 180 1 (A (B A) ↔ A = )
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ∖ cdif 3206   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551 This theorem is referenced by: (None)
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