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Theorem ssdifeq0 3299
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 3140 . . 3 (AA) = A
2 ssdifin0 3298 . . 3 (A ⊆ (BA) → (AA) = ∅)
31, 2syl5eqr 2083 . 2 (A ⊆ (BA) → A = ∅)
4 0ss 3249 . . 3 ∅ ⊆ (B ∖ ∅)
5 id 19 . . . 4 (A = ∅ → A = ∅)
6 difeq2 3050 . . . 4 (A = ∅ → (BA) = (B ∖ ∅))
75, 6sseq12d 2968 . . 3 (A = ∅ → (A ⊆ (BA) ↔ ∅ ⊆ (B ∖ ∅)))
84, 7mpbiri 157 . 2 (A = ∅ → A ⊆ (BA))
93, 8impbii 117 1 (A ⊆ (BA) ↔ A = ∅)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242  cdif 2908  cin 2910  wss 2911  c0 3218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219
This theorem is referenced by: (None)
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