Theorem ssexg 3866

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	⊢ ((A ⊆ B ∧ B ∈ 𝐶) → A ∈ V)

Proof of Theorem ssexg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 2940	. . . 4 ⊢ (x = B → (A ⊆ x ↔ A ⊆ B))
2	1	imbi1d 220	. . 3 ⊢ (x = B → ((A ⊆ x → A ∈ V) ↔ (A ⊆ B → A ∈ V)))
3		vex 2534	. . . 4 ⊢ x ∈ V
4	3	ssex 3864	. . 3 ⊢ (A ⊆ x → A ∈ V)
5	2, 4	vtoclg 2586	. 2 ⊢ (B ∈ 𝐶 → (A ⊆ B → A ∈ V))
6	5	impcom 116	1 ⊢ ((A ⊆ B ∧ B ∈ 𝐶) → A ∈ V)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1226   ∈ wcel 1370  Vcvv 2531   ⊆ wss 2890

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845

This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-in 2897  df-ss 2904

This theorem is referenced by:  ssexd  3867  difexg  3868  rabexg  3870  elssabg  3872  elpw2g  3880  abssexg  3904  snexgOLD  3905  snexg  3906  sess1  4038  sess2  4039  trsuc  4105  unexb  4123  uniexb  4151  xpexg  4375  riinint  4516  dmexg  4519  rnexg  4520  resexg  4573  resiexg  4576  imaexg  4603  exse2  4622  cnvexg  4778  coexg  4785  fabexg  4998  f1oabexg  5059  relrnfvex  5114  fvexg  5115  sefvex  5117  mptfvex  5177  mptexg  5307  ofres  5644  resfunexgALT  5656  cofunexg  5657  fnexALT  5659  f1dmex  5662  oprabexd  5673  mpt2exxg  5752  tposexg  5791  frecabex  5895  erex  6037