Theorem ssexd 3897

Description: A subclass of a set is a set. Deduction form of ssexg 3896. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
ssexd.1	⊢ (𝜑 → 𝐵 ∈ 𝐶)
ssexd.2	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
ssexd	⊢ (𝜑 → 𝐴 ∈ V)

Proof of Theorem ssexd

Step	Hyp	Ref	Expression
1		ssexd.2	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		ssexd.1	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
3		ssexg 3896	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
4	1, 2, 3	syl2anc 391	1 ⊢ (𝜑 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931

This theorem is referenced by:  fex2  5059  riotaexg  5472  opabbrex  5549  f1imaen2g  6273  genipv  6607  iseqss  9226  ovshftex  9420