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Theorem ssdif0im 3280
Description: Subclass implies empty difference. One direction of Exercise 7 of [TakeutiZaring] p. 22. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 2-Aug-2018.)
Assertion
Ref Expression
ssdif0im (AB → (AB) = ∅)

Proof of Theorem ssdif0im
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 imanim 784 . . . 4 ((x Ax B) → ¬ (x A ¬ x B))
2 eldif 2921 . . . 4 (x (AB) ↔ (x A ¬ x B))
31, 2sylnibr 601 . . 3 ((x Ax B) → ¬ x (AB))
43alimi 1341 . 2 (x(x Ax B) → x ¬ x (AB))
5 dfss2 2928 . 2 (ABx(x Ax B))
6 eq0 3233 . 2 ((AB) = ∅ ↔ x ¬ x (AB))
74, 5, 63imtr4i 190 1 (AB → (AB) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wal 1240   = wceq 1242   wcel 1390  cdif 2908  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-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219
This theorem is referenced by:  vdif0im  3281  difrab0eqim  3282  difid  3286  difin0  3291
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