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Theorem ssexg 3887
 Description: The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.)
Assertion
Ref Expression
ssexg ((AB B 𝐶) → A V)

Proof of Theorem ssexg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 sseq2 2961 . . . 4 (x = B → (AxAB))
21imbi1d 220 . . 3 (x = B → ((AxA V) ↔ (ABA V)))
3 vex 2554 . . . 4 x V
43ssex 3885 . . 3 (AxA V)
52, 4vtoclg 2607 . 2 (B 𝐶 → (ABA V))
65impcom 116 1 ((AB B 𝐶) → A V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ⊆ wss 2911 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-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-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866 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-in 2918  df-ss 2925 This theorem is referenced by:  ssexd  3888  difexg  3889  rabexg  3891  elssabg  3893  elpw2g  3901  abssexg  3925  snexgOLD  3926  snexg  3927  sess1  4059  sess2  4060  trsuc  4125  unexb  4143  uniexb  4171  xpexg  4395  riinint  4536  dmexg  4539  rnexg  4540  resexg  4593  resiexg  4596  imaexg  4623  exse2  4642  cnvexg  4798  coexg  4805  fabexg  5020  f1oabexg  5081  relrnfvex  5136  fvexg  5137  sefvex  5139  mptfvex  5199  mptexg  5329  ofres  5667  resfunexgALT  5679  cofunexg  5680  fnexALT  5682  f1dmex  5685  oprabexd  5696  mpt2exxg  5775  tposexg  5814  frecabex  5923  erex  6066  ssdomg  6194  fiprc  6228
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