Theorem ssexg 1702

Description: The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized).

Assertion

Ref	Expression
ssexg	⊢ (B ∈ C → (A ⊆ B → A ∈ V))

Proof of Theorem ssexg

Step	Hyp	Ref	Expression
1		sseq2 1522	. . 3 ⊢ (x = B → (A ⊆ x ↔ A ⊆ B))
2	1	imbi1d 465	. 2 ⊢ (x = B → ((A ⊆ x → A ∈ V) ↔ (A ⊆ B → A ∈ V)))
3		visset 1350	. . 3 ⊢ x ∈ V
4	3	ssex 1700	. 2 ⊢ (A ⊆ x → A ∈ V)
5	2, 4	vtoclg 1383	1 ⊢ (B ∈ C → (A ⊆ B → A ∈ V))

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487

This theorem is referenced by:  difexg 1703  rabexg 1705  elpw2g 1803  unexb 1950  difex2 1951  uniexb 1962  dmexg 2551  rnexg 2569  imaexg 2612  cnvexg 2669  coexg 2671  resfunexg 2717  fnex 2740  f1dmex 2819  tz7.48-3 2996  mapex 3261  ssdom2g 3312  pssnn 3428

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492

metamath.org