Theorem bdssexd 10696

Description: Bounded version of ssexd 3918. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdssexd.1	⊢ (𝜑 → 𝐵 ∈ 𝐶)
bdssexd.2	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
bdssexd.bd	⊢ BOUNDED 𝐴

Assertion

Ref	Expression
bdssexd	⊢ (𝜑 → 𝐴 ∈ V)

Proof of Theorem bdssexd

Step	Hyp	Ref	Expression
1		bdssexd.2	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		bdssexd.1	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
3		bdssexd.bd	. . 3 ⊢ BOUNDED 𝐴
4	3	bdssexg 10695	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
5	1, 2, 4	syl2anc 403	1 ⊢ (𝜑 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1433  Vcvv 2601   ⊆ wss 2973  BOUNDED wbdc 10631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bdsep 10675

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-bdc 10632

This theorem is referenced by: (None)