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Theorem ssdifeq0 3411
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 ⊆ (𝐵𝐴) ↔ 𝐴 = ∅)

Proof of Theorem ssdifeq0
StepHypRef Expression
1 inidm 3251 . . 3 (𝐴𝐴) = 𝐴
2 ssdifin0 3410 . . 3 (𝐴 ⊆ (𝐵𝐴) → (𝐴𝐴) = ∅)
31, 2syl5eqr 2161 . 2 (𝐴 ⊆ (𝐵𝐴) → 𝐴 = ∅)
4 0ss 3367 . . 3 ∅ ⊆ (𝐵 ∖ ∅)
5 id 19 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
6 difeq2 3154 . . . 4 (𝐴 = ∅ → (𝐵𝐴) = (𝐵 ∖ ∅))
75, 6sseq12d 3094 . . 3 (𝐴 = ∅ → (𝐴 ⊆ (𝐵𝐴) ↔ ∅ ⊆ (𝐵 ∖ ∅)))
84, 7mpbiri 167 . 2 (𝐴 = ∅ → 𝐴 ⊆ (𝐵𝐴))
93, 8impbii 125 1 (𝐴 ⊆ (𝐵𝐴) ↔ 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1314  cdif 3034  cin 3036  wss 3037  c0 3329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rab 2399  df-v 2659  df-dif 3039  df-in 3043  df-ss 3050  df-nul 3330
This theorem is referenced by: (None)
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